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A major challenge of AI + Science lies in their inherent incompatibility: today's AI is primarily based on connectionism, while science depends on symbolism. To bridge the two worlds, we propose a framework to seamlessly synergize Kolmogorov-Arnold Networks (KANs) and science. The framework highlights KANs' usage for three aspects of scientific discovery: identifying relevant features, revealing modular structures, and discovering symbolic formulas. The synergy is bidirectional: science to KAN (incorporating scientific knowledge into KANs), and KAN to science (extracting scientific insights from KANs). We highlight major new functionalities in the pykan package: (1) MultKAN: KANs with multiplication nodes. (2) kanpiler: a KAN compiler that compiles symbolic formulas into KANs. (3) tree converter: convert KANs (or any neural networks) to tree graphs. Based on these tools, we demonstrate KANs' capability to discover various types of physical laws, including conserved quantities, Lagrangians, symmetries, and constitutive laws., Comment: 27 pages, 14 figures

We demonstrate and investigate the remarkable robustness of Large Language Models by deleting and swapping adjacent layers. We find that deleting and swapping interventions retain 72-95\% of the original model's prediction accuracy without fine-tuning, whereas models with more layers exhibit more robustness. Based on the results of the layer-wise intervention and further experiments, we hypothesize the existence of four universal stages of inference across eight different models: detokenization, feature engineering, prediction ensembling, and residual sharpening. The first stage integrates local information, lifting raw token representations into higher-level contextual representations. Next is the iterative refinement of task and entity-specific features. Then, the second half of the model begins with a phase transition, where hidden representations align more with the vocabulary space due to specialized model components. Finally, the last layer sharpens the following token distribution by eliminating obsolete features that add noise to the prediction.

We introduce DafnyBench, the largest benchmark of its kind for training and evaluating machine learning systems for formal software verification. We test the ability of LLMs such as GPT-4 and Claude 3 to auto-generate enough hints for the Dafny formal verification engine to successfully verify over 750 programs with about 53,000 lines of code. The best model and prompting scheme achieved 68% success rate, and we quantify how this rate improves when retrying with error message feedback and how it deteriorates with the amount of required code and hints. We hope that DafnyBench will enable rapid improvements from this baseline as LLMs and verification techniques grow in quality., Comment: Code & dataset available at: https://github.com/sun-wendy/DafnyBench

When a neural network can learn multiple distinct algorithms to solve a task, how does it "choose" between them during training? To approach this question, we take inspiration from ecology: when multiple species coexist, they eventually reach an equilibrium where some survive while others die out. Analogously, we suggest that a neural network at initialization contains many solutions (representations and algorithms), which compete with each other under pressure from resource constraints, with the "fittest" ultimately prevailing. To investigate this Survival of the Fittest hypothesis, we conduct a case study on neural networks performing modular addition, and find that these networks' multiple circular representations at different Fourier frequencies undergo such competitive dynamics, with only a few circles surviving at the end. We find that the frequencies with high initial signals and gradients, the "fittest," are more likely to survive. By increasing the embedding dimension, we also observe more surviving frequencies. Inspired by the Lotka-Volterra equations describing the dynamics between species, we find that the dynamics of the circles can be nicely characterized by a set of linear differential equations. Our results with modular addition show that it is possible to decompose complicated representations into simpler components, along with their basic interactions, to offer insight on the training dynamics of representations.

Recent work has proposed the linear representation hypothesis: that language models perform computation by manipulating one-dimensional representations of concepts ("features") in activation space. In contrast, we explore whether some language model representations may be inherently multi-dimensional. We begin by developing a rigorous definition of irreducible multi-dimensional features based on whether they can be decomposed into either independent or non-co-occurring lower-dimensional features. Motivated by these definitions, we design a scalable method that uses sparse autoencoders to automatically find multi-dimensional features in GPT-2 and Mistral 7B. These auto-discovered features include strikingly interpretable examples, e.g. circular features representing days of the week and months of the year. We identify tasks where these exact circles are used to solve computational problems involving modular arithmetic in days of the week and months of the year. Finally, we provide evidence that these circular features are indeed the fundamental unit of computation in these tasks with intervention experiments on Mistral 7B and Llama 3 8B, and we find further circular representations by breaking down the hidden states for these tasks into interpretable components., Comment: Code and data at https://github.com/JoshEngels/MultiDimensionalFeatures

How do transformers model physics? Do transformers model systems with interpretable analytical solutions, or do they create "alien physics" that are difficult for humans to decipher? We take a step in demystifying this larger puzzle by investigating the simple harmonic oscillator (SHO), $\ddot{x}+2\gamma \dot{x}+\omega_0^2x=0$, one of the most fundamental systems in physics. Our goal is to identify the methods transformers use to model the SHO, and to do so we hypothesize and evaluate possible methods by analyzing the encoding of these methods' intermediates. We develop four criteria for the use of a method within the simple testbed of linear regression, where our method is $y = wx$ and our intermediate is $w$: (1) Can the intermediate be predicted from hidden states? (2) Is the intermediate's encoding quality correlated with model performance? (3) Can the majority of variance in hidden states be explained by the intermediate? (4) Can we intervene on hidden states to produce predictable outcomes? Armed with these two correlational (1,2), weak causal (3) and strong causal (4) criteria, we determine that transformers use known numerical methods to model trajectories of the simple harmonic oscillator, specifically the matrix exponential method. Our analysis framework can conveniently extend to high-dimensional linear systems and nonlinear systems, which we hope will help reveal the "world model" hidden in transformers., Comment: 9 pages, 9 figures

Ensuring that AI systems reliably and robustly avoid harmful or dangerous behaviours is a crucial challenge, especially for AI systems with a high degree of autonomy and general intelligence, or systems used in safety-critical contexts. In this paper, we will introduce and define a family of approaches to AI safety, which we will refer to as guaranteed safe (GS) AI. The core feature of these approaches is that they aim to produce AI systems which are equipped with high-assurance quantitative safety guarantees. This is achieved by the interplay of three core components: a world model (which provides a mathematical description of how the AI system affects the outside world), a safety specification (which is a mathematical description of what effects are acceptable), and a verifier (which provides an auditable proof certificate that the AI satisfies the safety specification relative to the world model). We outline a number of approaches for creating each of these three core components, describe the main technical challenges, and suggest a number of potential solutions to them. We also argue for the necessity of this approach to AI safety, and for the inadequacy of the main alternative approaches.

Integrable partial differential equation (PDE) systems are of great interest in natural science, but are exceedingly rare and difficult to discover. To solve this, we introduce OptPDE, a first-of-its-kind machine learning approach that Optimizes PDEs' coefficients to maximize their number of conserved quantities, $n_{\rm CQ}$, and thus discover new integrable systems. We discover four families of integrable PDEs, one of which was previously known, and three of which have at least one conserved quantity but are new to the literature to the best of our knowledge. We investigate more deeply the properties of one of these novel PDE families, $u_t = (u_x+a^2u_{xxx})^3$. Our paper offers a promising schema of AI-human collaboration for integrable system discovery: machine learning generates interpretable hypotheses for possible integrable systems, which human scientists can verify and analyze, to truly close the discovery loop.

Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs., Comment: 48 pages, 20 figures. Codes are available at https://github.com/KindXiaoming/pykan

We present GenEFT: an effective theory framework for shedding light on the statics and dynamics of neural network generalization, and illustrate it with graph learning examples. We first investigate the generalization phase transition as data size increases, comparing experimental results with information-theory-based approximations. We find generalization in a Goldilocks zone where the decoder is neither too weak nor too powerful. We then introduce an effective theory for the dynamics of representation learning, where latent-space representations are modeled as interacting particles (repons), and find that it explains our experimentally observed phase transition between generalization and overfitting as encoder and decoder learning rates are scanned. This highlights the power of physics-inspired effective theories for bridging the gap between theoretical predictions and practice in machine learning., Comment: 12 pages, 6 figures

Neural scaling laws characterize how model performance improves as the model size scales up. Inspired by empirical observations, we introduce a resource model of neural scaling. A task is usually composite hence can be decomposed into many subtasks, which compete for resources (measured by the number of neurons allocated to subtasks). On toy problems, we empirically find that: (1) The loss of a subtask is inversely proportional to its allocated neurons. (2) When multiple subtasks are present in a composite task, the resources acquired by each subtask uniformly grow as models get larger, keeping the ratios of acquired resources constants. We hypothesize these findings to be generally true and build a model to predict neural scaling laws for general composite tasks, which successfully replicates the neural scaling law of Chinchilla models reported in arXiv:2203.15556. We believe that the notion of resource used in this paper will be a useful tool for characterizing and diagnosing neural networks., Comment: 10 pages, 8 figures, Published as a workshop paper at ICLR 2024

We present MIPS, a novel method for program synthesis based on automated mechanistic interpretability of neural networks trained to perform the desired task, auto-distilling the learned algorithm into Python code. We test MIPS on a benchmark of 62 algorithmic tasks that can be learned by an RNN and find it highly complementary to GPT-4: MIPS solves 32 of them, including 13 that are not solved by GPT-4 (which also solves 30). MIPS uses an integer autoencoder to convert the RNN into a finite state machine, then applies Boolean or integer symbolic regression to capture the learned algorithm. As opposed to large language models, this program synthesis technique makes no use of (and is therefore not limited by) human training data such as algorithms and code from GitHub. We discuss opportunities and challenges for scaling up this approach to make machine-learned models more interpretable and trustworthy., Comment: 24 pages

External audits of AI systems are increasingly recognized as a key mechanism for AI governance. The effectiveness of an audit, however, depends on the degree of access granted to auditors. Recent audits of state-of-the-art AI systems have primarily relied on black-box access, in which auditors can only query the system and observe its outputs. However, white-box access to the system's inner workings (e.g., weights, activations, gradients) allows an auditor to perform stronger attacks, more thoroughly interpret models, and conduct fine-tuning. Meanwhile, outside-the-box access to training and deployment information (e.g., methodology, code, documentation, data, deployment details, findings from internal evaluations) allows auditors to scrutinize the development process and design more targeted evaluations. In this paper, we examine the limitations of black-box audits and the advantages of white- and outside-the-box audits. We also discuss technical, physical, and legal safeguards for performing these audits with minimal security risks. Given that different forms of access can lead to very different levels of evaluation, we conclude that (1) transparency regarding the access and methods used by auditors is necessary to properly interpret audit results, and (2) white- and outside-the-box access allow for substantially more scrutiny than black-box access alone., Comment: FAccT 2024

An essential goal in mechanistic interpretability to decode a network, i.e., to convert a neural network's raw weights to an interpretable algorithm. Given the difficulty of the decoding problem, progress has been made to understand the easier encoding problem, i.e., to convert an interpretable algorithm into network weights. Previous works focus on encoding existing algorithms into networks, which are interpretable by definition. However, focusing on encoding limits the possibility of discovering new algorithms that humans have never stumbled upon, but that are nevertheless interpretable. In this work, we explore the possibility of using hypernetworks to generate interpretable networks whose underlying algorithms are not yet known. The hypernetwork is carefully designed such that it can control network complexity, leading to a diverse family of interpretable algorithms ranked by their complexity. All of them are interpretable in hindsight, although some of them are less intuitive to humans, hence providing new insights regarding how to "think" like a neural network. For the task of computing L1 norms, hypernetworks find three algorithms: (a) the double-sided algorithm, (b) the convexity algorithm, (c) the pudding algorithm, although only the first algorithm was expected by the authors before experiments. We automatically classify these algorithms and analyze how these algorithmic phases develop during training, as well as how they are affected by complexity control. Furthermore, we show that a trained hypernetwork can correctly construct models for input dimensions not seen in training, demonstrating systematic generalization., Comment: 15 pages, 7 figures

Recurrent neural networks (RNNs) trained on compositional tasks can exhibit functional modularity, in which neurons can be clustered by activity similarity and participation in shared computational subtasks. Unlike brains, these RNNs do not exhibit anatomical modularity, in which functional clustering is correlated with strong recurrent coupling and spatial localization of functional clusters. Contrasting with functional modularity, which can be ephemerally dependent on the input, anatomically modular networks form a robust substrate for solving the same subtasks in the future. To examine whether it is possible to grow brain-like anatomical modularity, we apply a recent machine learning method, brain-inspired modular training (BIMT), to a network being trained to solve a set of compositional cognitive tasks. We find that functional and anatomical clustering emerge together, such that functionally similar neurons also become spatially localized and interconnected. Moreover, compared to standard $L_1$ or no regularization settings, the model exhibits superior performance by optimally balancing task performance and network sparsity. In addition to achieving brain-like organization in RNNs, our findings also suggest that BIMT holds promise for applications in neuromorphic computing and enhancing the interpretability of neural network architectures., Comment: 8 pages, 6 figures

Large Language Models (LLMs) have impressive capabilities, but are prone to outputting falsehoods. Recent work has developed techniques for inferring whether a LLM is telling the truth by training probes on the LLM's internal activations. However, this line of work is controversial, with some authors pointing out failures of these probes to generalize in basic ways, among other conceptual issues. In this work, we use high-quality datasets of simple true/false statements to study in detail the structure of LLM representations of truth, drawing on three lines of evidence: 1. Visualizations of LLM true/false statement representations, which reveal clear linear structure. 2. Transfer experiments in which probes trained on one dataset generalize to different datasets. 3. Causal evidence obtained by surgically intervening in a LLM's forward pass, causing it to treat false statements as true and vice versa. Overall, we present evidence that at sufficient scale, LLMs linearly represent the truth or falsehood of factual statements. We also show that simple difference-in-mean probes generalize as well as other probing techniques while identifying directions which are more causally implicated in model outputs., Comment: Conference on Language Modeling, 2024

AI companies are attempting to create AI systems that outperform humans at most economically valuable work. Current AI models are already automating away the livelihoods of some artists, actors, and writers. But there is infighting between those who prioritize current harms and future harms. We construct a game-theoretic model of conflict to study the causes and consequences of this disunity. Our model also helps explain why throughout history, stakeholders sharing a common threat have found it advantageous to unite against it, and why the common threat has in turn found it advantageous to divide and conquer. Under realistic parameter assumptions, our model makes several predictions that find preliminary corroboration in the historical-empirical record. First, current victims of AI-driven disempowerment need the future victims to realize that their interests are also under serious and imminent threat, so that future victims are incentivized to support current victims in solidarity. Second, the movement against AI-driven disempowerment can become more united, and thereby more likely to prevail, if members believe that their efforts will be successful as opposed to futile. Finally, the movement can better unite and prevail if its members are less myopic. Myopic members prioritize their future well-being less than their present well-being, and are thus disinclined to solidarily support current victims today at personal cost, even if this is necessary to counter the shared threat of AI-driven disempowerment., Comment: 28 pages, nine visualizations (seven figures and two tables)

We attribute grokking, the phenomenon where generalization is much delayed after memorization, to compression. To do so, we define linear mapping number (LMN) to measure network complexity, which is a generalized version of linear region number for ReLU networks. LMN can nicely characterize neural network compression before generalization. Although the $L_2$ norm has been a popular choice for characterizing model complexity, we argue in favor of LMN for a number of reasons: (1) LMN can be naturally interpreted as information/computation, while $L_2$ cannot. (2) In the compression phase, LMN has linear relations with test losses, while $L_2$ is correlated with test losses in a complicated nonlinear way. (3) LMN also reveals an intriguing phenomenon of the XOR network switching between two generalization solutions, while $L_2$ does not. Besides explaining grokking, we argue that LMN is a promising candidate as the neural network version of the Kolmogorov complexity since it explicitly considers local or conditioned linear computations aligned with the nature of modern artificial neural networks.

Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as $N^{-\alpha}$, $\alpha=4/d$, where $N$ is the number of model parameters, and $d$ is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem $y=x^2$ manifests a different scaling law ($\alpha=1$) from their predictions ($\alpha=4$). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the $N^{-1}$ scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning., Comment: 14 pages, 13 figures. Note from authors: the theory in this paper is questionable; we are trying our best to fix it. Empirical results still stand

The capabilities of large language models (LLMs) have sparked debate over whether such systems just learn an enormous collection of superficial statistics or a set of more coherent and grounded representations that reflect the real world. We find evidence for the latter by analyzing the learned representations of three spatial datasets (world, US, NYC places) and three temporal datasets (historical figures, artworks, news headlines) in the Llama-2 family of models. We discover that LLMs learn linear representations of space and time across multiple scales. These representations are robust to prompting variations and unified across different entity types (e.g. cities and landmarks). In addition, we identify individual "space neurons" and "time neurons" that reliably encode spatial and temporal coordinates. While further investigation is needed, our results suggest modern LLMs learn rich spatiotemporal representations of the real world and possess basic ingredients of a world model.

We describe a path to humanity safely thriving with powerful Artificial General Intelligences (AGIs) by building them to provably satisfy human-specified requirements. We argue that this will soon be technically feasible using advanced AI for formal verification and mechanistic interpretability. We further argue that it is the only path which guarantees safe controlled AGI. We end with a list of challenge problems whose solution would contribute to this positive outcome and invite readers to join in this work., Comment: 17 pages

Do neural networks, trained on well-understood algorithmic tasks, reliably rediscover known algorithms for solving those tasks? Several recent studies, on tasks ranging from group arithmetic to in-context linear regression, have suggested that the answer is yes. Using modular addition as a prototypical problem, we show that algorithm discovery in neural networks is sometimes more complex. Small changes to model hyperparameters and initializations can induce the discovery of qualitatively different algorithms from a fixed training set, and even parallel implementations of multiple such algorithms. Some networks trained to perform modular addition implement a familiar Clock algorithm; others implement a previously undescribed, less intuitive, but comprehensible procedure which we term the Pizza algorithm, or a variety of even more complex procedures. Our results show that even simple learning problems can admit a surprising diversity of solutions, motivating the development of new tools for characterizing the behavior of neural networks across their algorithmic phase space., Comment: Accepted by NeurIPS 2023

Discovering conservation laws for a given dynamical system is important but challenging. In a theorist setup (differential equations and basis functions are both known), we propose the Sparse Invariant Detector (SID), an algorithm that auto-discovers conservation laws from differential equations. Its algorithmic simplicity allows robustness and interpretability of the discovered conserved quantities. We show that SID is able to rediscover known and even discover new conservation laws in a variety of systems. For two examples in fluid mechanics and atmospheric chemistry, SID discovers 14 and 3 conserved quantities, respectively, where only 12 and 2 were previously known to domain experts., Comment: The codes are available here: https://github.com/KindXiaoming/sid

We introduce Brain-Inspired Modular Training (BIMT), a method for making neural networks more modular and interpretable. Inspired by brains, BIMT embeds neurons in a geometric space and augments the loss function with a cost proportional to the length of each neuron connection. We demonstrate that BIMT discovers useful modular neural networks for many simple tasks, revealing compositional structures in symbolic formulas, interpretable decision boundaries and features for classification, and mathematical structure in algorithmic datasets. The ability to directly see modules with the naked eye can complement current mechanistic interpretability strategies such as probes, interventions or staring at all weights., Comment: Codes are available here: https://github.com/KindXiaoming/BIMT

Since diffusion models (DM) and the more recent Poisson flow generative models (PFGM) are inspired by physical processes, it is reasonable to ask: Can physical processes offer additional new generative models? We show that the answer is yes. We introduce a general family, Generative Models from Physical Processes (GenPhys), where we translate partial differential equations (PDEs) describing physical processes to generative models. We show that generative models can be constructed from s-generative PDEs (s for smooth). GenPhys subsume the two existing generative models (DM and PFGM) and even give rise to new families of generative models, e.g., "Yukawa Generative Models" inspired from weak interactions. On the other hand, some physical processes by default do not belong to the GenPhys family, e.g., the wave equation and the Schr\"{o}dinger equation, but could be made into the GenPhys family with some modifications. Our goal with GenPhys is to explore and expand the design space of generative models.

We propose the Quantization Model of neural scaling laws, explaining both the observed power law dropoff of loss with model and data size, and also the sudden emergence of new capabilities with scale. We derive this model from what we call the Quantization Hypothesis, where network knowledge and skills are "quantized" into discrete chunks ($\textbf{quanta}$). We show that when quanta are learned in order of decreasing use frequency, then a power law in use frequencies explains observed power law scaling of loss. We validate this prediction on toy datasets, then study how scaling curves decompose for large language models. Using language model gradients, we automatically decompose model behavior into a diverse set of skills (quanta). We tentatively find that the frequency at which these quanta are used in the training distribution roughly follows a power law corresponding with the empirical scaling exponent for language models, a prediction of our theory., Comment: 24 pages, 18 figures, NeurIPS 2023

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for $N$ dimensional data by embedding paths in $N{+}D$ dimensional space while still controlling the progression with a simple scalar norm of the $D$ additional variables. The new models reduce to PFGM when $D{=}1$ and to diffusion models when $D{\to}\infty$. The flexibility of choosing $D$ allows us to trade off robustness against rigidity as increasing $D$ results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of $D$, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models ($D{\to} \infty$) to any finite $D$ values. Our experiments show that models with finite $D$ can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ $64{\times}64$ datasets, with FID scores of $1.91/2.43$ when $D{=}2048/128$. In class-conditional setting, $D{=}2048$ yields current state-of-the-art FID of $1.74$ on CIFAR-10. In addition, we demonstrate that models with smaller $D$ exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp, Comment: Code is available at https://github.com/Newbeeer/pfgmpp

We explore unique considerations involved in fitting ML models to data with very high precision, as is often required for science applications. We empirically compare various function approximation methods and study how they scale with increasing parameters and data. We find that neural networks can often outperform classical approximation methods on high-dimensional examples, by auto-discovering and exploiting modular structures therein. However, neural networks trained with common optimizers are less powerful for low-dimensional cases, which motivates us to study the unique properties of neural network loss landscapes and the corresponding optimization challenges that arise in the high precision regime. To address the optimization issue in low dimensions, we develop training tricks which enable us to train neural networks to extremely low loss, close to the limits allowed by numerical precision.

Grokking, the unusual phenomenon for algorithmic datasets where generalization happens long after overfitting the training data, has remained elusive. We aim to understand grokking by analyzing the loss landscapes of neural networks, identifying the mismatch between training and test losses as the cause for grokking. We refer to this as the "LU mechanism" because training and test losses (against model weight norm) typically resemble "L" and "U", respectively. This simple mechanism can nicely explain many aspects of grokking: data size dependence, weight decay dependence, the emergence of representations, etc. Guided by the intuitive picture, we are able to induce grokking on tasks involving images, language and molecules. In the reverse direction, we are able to eliminate grokking for algorithmic datasets. We attribute the dramatic nature of grokking for algorithmic datasets to representation learning.

We propose a new "Poisson flow" generative model (PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the $z=0$ hyperplane in a space augmented with an additional dimension $z$, generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the $z=0$ plane transforms into a distribution on the hemisphere of radius $r$ that becomes uniform in the $r \to\infty$ limit. To learn the bijective transformation, we estimate the normalized field in the augmented space. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the unaugmented data manifold when the $z$ reaches zero. Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of $9.68$ and a FID score of $2.35$. It also performs on par with the state-of-the-art SDE approaches while offering $10\times $ to $20 \times$ acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method. The code is available at https://github.com/Newbeeer/poisson_flow ., Comment: Accepted by NeurIPS 2022

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a "Goldilocks zone" (including comprehension and grokking) between memorization and confusion. We find on transformers the grokking phase stays closer to the memorization phase (compared to the comprehension phase), leading to delayed generalization. The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning., Comment: Accepted by NeurIPS 2022

At the heart of both lossy compression and clustering is a trade-off between the fidelity and size of the learned representation. Our goal is to map out and study the Pareto frontier that quantifies this trade-off. We focus on the optimization of the Deterministic Information Bottleneck (DIB) objective over the space of hard clusterings. To this end, we introduce the primal DIB problem, which we show results in a much richer frontier than its previously studied Lagrangian relaxation when optimized over discrete search spaces. We present an algorithm for mapping out the Pareto frontier of the primal DIB trade-off that is also applicable to other two-objective clustering problems. We study general properties of the Pareto frontier, and we give both analytic and numerical evidence for logarithmic sparsity of the frontier in general. We provide evidence that our algorithm has polynomial scaling despite the super-exponential search space, and additionally, we propose a modification to the algorithm that can be used where sampling noise is expected to be significant. Finally, we use our algorithm to map the DIB frontier of three different tasks: compressing the English alphabet, extracting informative color classes from natural images, and compressing a group theory-inspired dataset, revealing interesting features of frontier, and demonstrating how the structure of the frontier can be used for model selection with a focus on points previously hidden by the cloak of the convex hull., Comment: 28 pages, 12 figures

We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a non-linear generalization of linear independence). Our independence module can be viewed as a nonlinear generalization of singular value decomposition. Our method can readily handle inductive biases for conservation laws. We validate it with examples including the 3-body problem, the KdV equation and nonlinear Schr\"odinger equation., Comment: 15 pages, 12 figures

It has been an open question in deep learning if fault-tolerant computation is possible: can arbitrarily reliable computation be achieved using only unreliable neurons? In the grid cells of the mammalian cortex, analog error correction codes have been observed to protect states against neural spiking noise, but their role in information processing is unclear. Here, we use these biological error correction codes to develop a universal fault-tolerant neural network that achieves reliable computation if the faultiness of each neuron lies below a sharp threshold; remarkably, we find that noisy biological neurons fall below this threshold. The discovery of a phase transition from faulty to fault-tolerant neural computation suggests a mechanism for reliable computation in the cortex and opens a path towards understanding noisy analog systems relevant to artificial intelligence and neuromorphic computing.

Integrating physical inductive biases into machine learning can improve model generalizability. We generalize the successful paradigm of physics-informed learning (PIL) into a more general framework that also includes what we term physics-augmented learning (PAL). PIL and PAL complement each other by handling discriminative and generative properties, respectively. In numerical experiments, we show that PAL performs well on examples where PIL is inapplicable or inefficient., Comment: 10 pages, 3 figures, 4 tables

We present an automated method for finding hidden symmetries, defined as symmetries that become manifest only in a new coordinate system that must be discovered. Its core idea is to quantify asymmetry as violation of certain partial differential equations, and to numerically minimize such violation over the space of all invertible transformations, parametrized as invertible neural networks. For example, our method rediscovers the famous Gullstrand-Painleve metric that manifests hidden translational symmetry in the Schwarzschild metric of non-rotating black holes, as well as Hamiltonicity, modularity and other simplifying traits not traditionally viewed as symmetries., Comment: Replaced to match accepted PRL version. Improved training, discussion & noise modeling. 14 pages & 4 figs including supplementary material

We present an automated method for measuring media bias. Inferring which newspaper published a given article, based only on the frequencies with which it uses different phrases, leads to a conditional probability distribution whose analysis lets us automatically map newspapers and phrases into a bias space. By analyzing roughly a million articles from roughly a hundred newspapers for bias in dozens of news topics, our method maps newspapers into a two-dimensional bias landscape that agrees well with previous bias classifications based on human judgement. One dimension can be interpreted as traditional left-right bias, the other as establishment bias. This means that although news bias is inherently political, its measurement need not be., Comment: 29 pages, 23 figs; data available at https://space.mit.edu/home/tegmark/phrasebias.html

Energy conservation is a basic physics principle, the breakdown of which often implies new physics. This paper presents a method for data-driven "new physics" discovery. Specifically, given a trajectory governed by unknown forces, our Neural New-Physics Detector (NNPhD) aims to detect new physics by decomposing the force field into conservative and non-conservative components, which are represented by a Lagrangian Neural Network (LNN) and a universal approximator network (UAN), respectively, trained to minimize the force recovery error plus a constant $\lambda$ times the magnitude of the predicted non-conservative force. We show that a phase transition occurs at $\lambda$=1, universally for arbitrary forces. We demonstrate that NNPhD successfully discovers new physics in toy numerical experiments, rediscovering friction (1493) from a damped double pendulum, Neptune from Uranus' orbit (1846) and gravitational waves (2017) from an inspiraling orbit. We also show how NNPhD coupled with an integrator outperforms previous methods for predicting the future of a damped double pendulum., Comment: 17 pages, 7 figs, 2 tables; typo correction

Precision calibration poses challenges to experiments probing the redshifted 21-cm signal of neutral hydrogen from the Cosmic Dawn and Epoch of Reionization (z~30-6). In both interferometric and global signal experiments, systematic calibration is the leading source of error. Though many aspects of calibration have been studied, the overlap between the two types of instruments has received less attention. We investigate the sky based calibration of total power measurements with a HERA dish and an EDGES style antenna to understand the role of auto-correlations in the calibration of an interferometer and the role of sky in calibrating a total power instrument. Using simulations we study various scenarios such as time variable gain, incomplete sky calibration model, and primary beam model. We find that temporal gain drifts, sky model incompleteness, and beam inaccuracies cause biases in the receiver gain amplitude and the receiver temperature estimates. In some cases, these biases mix spectral structure between beam and sky resulting in spectrally variable gain errors. Applying the calibration method to the HERA and EDGES data, we find good agreement with calibration via the more standard methods. Although instrumental gains are consistent with beam and sky errors similar in scale to those simulated, the receiver temperatures show significant deviations from expected values. While we show that it is possible to partially mitigate biases due to model inaccuracies by incorporating a time-dependent gain model in calibration, the resulting errors on calibration products are larger and more correlated. Completely addressing these biases will require more accurate sky and primary beam models., Comment: 16 pages, 13 figures, 1 table; accepted for publication in MNRAS main journal

We present AI Poincar\'e, a machine learning algorithm for auto-discovering conserved quantities using trajectory data from unknown dynamical systems. We test it on five Hamiltonian systems, including the gravitational 3-body problem, and find that it discovers not only all exactly conserved quantities, but also periodic orbits, phase transitions and breakdown timescales for approximate conservation laws., Comment: Replaced by accepted PRL version; expanded validation, improved presentation, more legible figs

We present an improved method for symbolic regression that seeks to fit data to formulas that are Pareto-optimal, in the sense of having the best accuracy for a given complexity. It improves on the previous state-of-the-art by typically being orders of magnitude more robust toward noise and bad data, and also by discovering many formulas that stumped previous methods. We develop a method for discovering generalized symmetries (arbitrary modularity in the computational graph of a formula) from gradient properties of a neural network fit. We use normalizing flows to generalize our symbolic regression method to probability distributions from which we only have samples, and employ statistical hypothesis testing to accelerate robust brute-force search., Comment: 17 pages, 6 figs, replaced to match accepted NeurIPS version

We present a method for unsupervised learning of equations of motion for objects in raw and optionally distorted unlabeled video. We first train an autoencoder that maps each video frame into a low-dimensional latent space where the laws of motion are as simple as possible, by minimizing a combination of non-linearity, acceleration and prediction error. Differential equations describing the motion are then discovered using Pareto-optimal symbolic regression. We find that our pre-regression ("pregression") step is able to rediscover Cartesian coordinates of unlabeled moving objects even when the video is distorted by a generalized lens. Using intuition from multidimensional knot-theory, we find that the pregression step is facilitated by first adding extra latent space dimensions to avoid topological problems during training and then removing these extra dimensions via principal component analysis., Comment: Expanded and improved physics discussion, additional method details. 9 pages, 7 figs

The key challenge in the observation of the redshifted 21-cm signal from cosmic reionization is its separation from the much brighter foreground emission. Such separation relies on the different spectral properties of the two components, although, in real life, the foreground intrinsic spectrum is often corrupted by the instrumental response, inducing systematic effects that can further jeopardize the measurement of the 21-cm signal. In this paper, we use Gaussian Process Regression to model both foreground emission and instrumental systematics in $\sim 2$ hours of data from the Hydrogen Epoch of Reionization Array. We find that a simple co-variance model with three components matches the data well, giving a residual power spectrum with white noise properties. These consist of an "intrinsic" and instrumentally corrupted component with a coherence-scale of 20 MHz and 2.4 MHz respectively (dominating the line of sight power spectrum over scales $k_{\parallel} \le 0.2$ h cMpc$^{-1}$) and a baseline dependent periodic signal with a period of $\sim 1$ MHz (dominating over $k_{\parallel} \sim 0.4 - 0.8$h cMpc$^{-1}$) which should be distinguishable from the 21-cm EoR signal whose typical coherence-scales is $\sim 0.8$ MHz., Comment: 15 pages, 15 figures, 1 table, Accepted to MNRAS

In 21 cm cosmology, precision calibration is key to the separation of the neutral hydrogen signal from very bright but spectrally-smooth astrophysical foregrounds. The Hydrogen Epoch of Reionization Array (HERA), an interferometer specialized for 21 cm cosmology and now under construction in South Africa, was designed to be largely calibrated using the self-consistency of repeated measurements of the same interferometric modes. This technique, known as "redundant-baseline calibration" resolves most of the internal degrees of freedom in the calibration problem. It assumes, however, on antenna elements with identical primary beams placed precisely on a redundant grid. In this work, we review the detailed implementation of the algorithms enabling redundant-baseline calibration and report results with HERA data. We quantify the effects of real-world non-redundancy and how they compare to the idealized scenario in which redundant measurements differ only in their noise realizations. Finally, we study how non-redundancy can produce spurious temporal structure in our calibration solutions--both in data and in simulations--and present strategies for mitigating that structure., Comment: 24 Pages, 19 Figures. Updated to match the accepted MNRAS version

The goal of lossy data compression is to reduce the storage cost of a data set $X$ while retaining as much information as possible about something ($Y$) that you care about. For example, what aspects of an image $X$ contain the most information about whether it depicts a cat? Mathematically, this corresponds to finding a mapping $X\to Z\equiv f(X)$ that maximizes the mutual information $I(Z,Y)$ while the entropy $H(Z)$ is kept below some fixed threshold. We present a method for mapping out the Pareto frontier for classification tasks, reflecting the tradeoff between retained entropy and class information. We first show how a random variable $X$ (an image, say) drawn from a class $Y\in\{1,...,n\}$ can be distilled into a vector $W=f(X)\in \mathbb{R}^{n-1}$ losslessly, so that $I(W,Y)=I(X,Y)$; for example, for a binary classification task of cats and dogs, each image $X$ is mapped into a single real number $W$ retaining all information that helps distinguish cats from dogs. For the $n=2$ case of binary classification, we then show how $W$ can be further compressed into a discrete variable $Z=g_\beta(W)\in\{1,...,m_\beta\}$ by binning $W$ into $m_\beta$ bins, in such a way that varying the parameter $\beta$ sweeps out the full Pareto frontier, solving a generalization of the Discrete Information Bottleneck (DIB) problem. We argue that the most interesting points on this frontier are "corners" maximizing $I(Z,Y)$ for a fixed number of bins $m=2,3...$ which can be conveniently be found without multiobjective optimization. We apply this method to the CIFAR-10, MNIST and Fashion-MNIST datasets, illustrating how it can be interpreted as an information-theoretically optimal image clustering algorithm., Comment: Replaced to match version published in Entropy. 17 pages, 9 figs; improved discussion, comparison with Blahut-Arimoto method

The Information Bottleneck (IB) method (\cite{tishby2000information}) provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective $I(X;Z)-\beta I(Y;Z)$ employs a Lagrange multiplier $\beta$ to tune this trade-off. However, in practice, not only is $\beta$ chosen empirically without theoretical guidance, there is also a lack of theoretical understanding between $\beta$, learnability, the intrinsic nature of the dataset and model capacity. In this paper, we show that if $\beta$ is improperly chosen, learning cannot happen -- the trivial representation $P(Z|X)=P(Z)$ becomes the global minimum of the IB objective. We show how this can be avoided, by identifying a sharp phase transition between the unlearnable and the learnable which arises as $\beta$ is varied. This phase transition defines the concept of IB-Learnability. We prove several sufficient conditions for IB-Learnability, which provides theoretical guidance for choosing a good $\beta$. We further show that IB-learnability is determined by the largest confident, typical, and imbalanced subset of the examples (the conspicuous subset), and discuss its relation with model capacity. We give practical algorithms to estimate the minimum $\beta$ for a given dataset. We also empirically demonstrate our theoretical conditions with analyses of synthetic datasets, MNIST, and CIFAR10., Comment: Accepted at UAI 2019

A core challenge for both physics and artificial intellicence (AI) is symbolic regression: finding a symbolic expression that matches data from an unknown function. Although this problem is likely to be NP-hard in principle, functions of practical interest often exhibit symmetries, separability, compositionality and other simplifying properties. In this spirit, we develop a recursive multidimensional symbolic regression algorithm that combines neural network fitting with a suite of physics-inspired techniques. We apply it to 100 equations from the Feynman Lectures on Physics, and it discovers all of them, while previous publicly available software cracks only 71; for a more difficult test set, we improve the state of the art success rate from 15% to 90%., Comment: 15 pages, 2 figs. Our code is available at https://github.com/SJ001/AI-Feynman and our Feynman Symbolic Regression Database for benchmarking can be downloaded at https://space.mit.edu/home/tegmark/aifeynman.html

The emergence of artificial intelligence (AI) and its progressively wider impact on many sectors across the society requires an assessment of its effect on sustainable development. Here we analyze published evidence of positive or negative impacts of AI on the achievement of each of the 17 goals and 169 targets of the 2030 Agenda for Sustainable Development. We find that AI can support the achievement of 128 targets across all SDGs, but it may also inhibit 58 targets. Notably, AI enables new technologies that improve efficiency and productivity, but it may also lead to increased inequalities among and within countries, thus hindering the achievement of the 2030 Agenda. The fast development of AI needs to be supported by appropriate policy and regulation. Otherwise, it would lead to gaps in transparency, accountability, safety and ethical standards of AI-based technology, which could be detrimental towards the development and sustainable use of AI. Finally, there is a lack of research assessing the medium- and long-term impacts of AI. It is therefore essential to reinforce the global debate regarding the use of AI and to develop the necessary regulatory insight and oversight for AI-based technologies.