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We present a new method for enhancing symbolic regression for differential equations via dimensional analysis, specifically Ipsen's and Buckingham pi methods. Since symbolic regression often suffers from high computational costs and overfitting, non-dimensionalizing datasets reduces the number of input variables, simplifies the search space, and ensures that derived equations are physically meaningful. As our main contribution, we integrate Ipsen's method of dimensional analysis with Universal Physics-Informed Neural Networks. We also combine dimensional analysis with the AI Feynman symbolic regression algorithm to show that dimensional analysis significantly improves the accuracy of the recovered equation. The results demonstrate that transforming data into a dimensionless form significantly decreases computation time and improves accuracy of the recovered hidden term. For algebraic equations, using the Buckingham pi theorem reduced complexity, allowing the AI Feynman model to converge faster with fewer data points and lower error rates. For differential equations, Ipsen's method was combined with Universal Physics-Informed Neural Networks (UPINNs) to identify hidden terms more effectively. These findings suggest that integrating dimensional analysis with symbolic regression can significantly lower computational costs, enhance model interpretability, and increase accuracy, providing a robust framework for automated discovery of governing equations in complex systems when data is limited.

We revisit the problem of a charged particle scattering off of an Aharonov-Bohm cosmic string. A classic computation gave an infinite total scattering cross section, leading to a Callan-Rubakov-like enhancement which can have important implications on baryon number asymmetry in the early universe. However, unlike the Callan-Rubakov effect, the Aharonov-Bohm interaction is topological and thus it is surprising that it leads to such a dramatic dynamical effect for single particle scattering. We reexamine this old problem through the modern lens of generalized global symmetries by embedding Aharanov-Bohm strings in a discrete gauge theory. We show that the scattering cross section is suppressed by the core size and there is thus no Callan-Rubakov-like enhancement., Comment: 5 pages

Accurately quantifying uncertainty in large language models (LLMs) is crucial for their reliable deployment, especially in high-stakes applications. Current state-of-the-art methods for measuring semantic uncertainty in LLMs rely on strict bidirectional entailment criteria between multiple generated responses and also depend on sequence likelihoods. While effective, these approaches often overestimate uncertainty due to their sensitivity to minor wording differences, additional correct information, and non-important words in the sequence. We propose a novel approach that leverages semantic embeddings to achieve smoother and more robust estimation of semantic uncertainty in LLMs. By capturing semantic similarities without depending on sequence likelihoods, our method inherently reduces any biases introduced by irrelevant words in the answers. Furthermore, we introduce an amortised version of our approach by explicitly modelling semantics as latent variables in a joint probabilistic model. This allows for uncertainty estimation in the embedding space with a single forward pass, significantly reducing computational overhead compared to existing multi-pass methods. Experiments across multiple question-answering datasets and frontier LLMs demonstrate that our embedding-based methods provide more accurate and nuanced uncertainty quantification than traditional approaches.

Free-induction-decay (FID) magnetometers have evolved as simple magnetic sensors for sensitive detection of unknown magnetic fields. However, these magnetometers suffer from a fundamental problem known as a "dead zone," making them insensitive to certain magnetic field directions. Here, we demonstrate a simple experimental scheme for the dead-zone-free operation of a FID atomic magnetometer. Using a single laser beam containing equal strength of linear- and circular-polarization components and amplitude-modulation at a low-duty cycle, we have synchronously pumped the rubidium-87 atoms with both first- and second-order frequency harmonics. Such a novel pumping scheme has enabled us to observe the free Larmor precession of atomic spins at a frequency of $\Omega_L$ (orientation) and/or 2$\Omega_L$ (alignment) in a single FID signal, depending on the direction of the external magnetic field. We observed that the amplitude of the FID signal does not go to zero for any magnetic field direction, proving the absence of dead zones in the magnetometer. The magnetometer has a sensitivity in the range of 3.5 - 7.8 pT/$\sqrt{Hz}$ in all directions. Such a generic experimental scheme can play a crucial role in developing miniaturized atomic magnetometers for various practical applications, including geomagnetic applications., Comment: 5 pages, 4 figures

How and where the first generation of inner solar system planetesimals formed remains poorly understood. Potential formation regions are the silicate condensation line and water-snowline of the solar protoplanetary disk. Whether the chemical compositions of these planetesimals align with accretion at the silicate condensation line (water-free and reduced) or water-snowline (water-bearing and oxidized) is, however, unknown. Here we use Fe/Ni and Fe/Co ratios of magmatic iron meteorites to quantify the oxidation states of the earliest planetesimals associated with non-carbonaceous (NC) and carbonaceous (CC) reservoirs, representing the inner and outer solar system, respectively. Our results show that the earliest NC planetesimals contained substantial amounts of oxidized Fe in their mantles (3-19 wt% FeO). In turn, we argue that this required the accretion of water-bearing materials into these NC planetesimals. The presence of substantial quantities of moderately and highly volatile elements in their parent cores is also inconsistent with their accretion at the silicate condensation line and favors instead their formation at or beyond the water-snowline. Similar oxidation states in the early-formed parent bodies of NC iron meteorites and those of NC achondrites and chondrites with diverse accretion ages suggests that the formation of oxidized planetesimals from water-bearing materials was widespread in the early history of the inner solar system., Comment: 28 pages, 3 main figures, 6 Extended data figures, 4 Extended data tables

Kumar used a switching lemma to prove exponential-size lower bounds for a circuit class GC^0 that not only contains AC^0 but can--with a single gate--compute functions that require exponential-size TC^0 circuits. His main result was that switching-lemma lower bounds for AC^0 lift to GC^0 with no loss in parameters, even though GC^0 requires exponential-size TC^0 circuits. Informally, GC^0 is AC^0 with unbounded-fan-in gates that behave arbitrarily inside a sufficiently small Hamming ball but must be constant outside it. We show that polynomial-method lower bounds for AC^0[p] lift to GC^0[p] with no loss in parameters, complementing Kumar's result for GC^0 and the switching lemma. As an application, we prove Majority requires depth-d GC^0[p] circuits of size $2^{\Omega(n^{1/2(d-1)})}$, matching the state-of-the-art lower bounds for AC^0[p]. We also show that E^NP requires exponential-size GCC^0 circuits (the union of GC^0[m] for all m), extending the result of Williams. It is striking that the switching lemma, polynomial method, and algorithmic method all generalize to GC^0-related classes, with the first two methods doing so without any loss. We also establish the strongest known unconditional separations between quantum and classical circuits: 1. There's an oracle relative to which BQP is not contained in the class of languages decidable by uniform families of size-$2^{n^{O(1)}}$ GC^0 circuits, generalizing Raz and Tal's relativized separation of BQP from the polynomial hierarchy. 2. There's a search problem that QNC^0 circuits can solve but average-case hard for exponential-size GC^0 circuits. 3. There's a search problem that QNC^0/qpoly circuits can solve but average-case hard for exponential-size GC^0[p] circuits. 4. There's an interactive problem that QNC^0 circuits can solve but exponential-size GC^0[p] circuits cannot., Comment: 48 pages; v2: title change and minor updates to the introduction

Those experiencing strokes, traumatic brain injuries, and drug complications can often end up hospitalized and diagnosed with coma or locked-in syndrome. Such mental impediments can permanently alter the neurological pathways in work and significantly decrease the quality of life (QoL). It is critical to translate brain signals into images to gain a deeper understanding of the thoughts of a comatose patient. Traditionally, brain signals collected by an EEG could only be translated into text, but with the novel method of an open-source model available on GitHub, DreamDiffusion can be used to convert brain waves into images directly. DreamDiffusion works by extracting features from EEG signals and then using the features to create images through StableDiffusion. Upon this, we made further improvements that could make StableDiffusion the forerunner technology in waves to media translation. In our study, we begin by modifying the existing DreamDiffusion codebase so that it does not require any prior setup, avoiding any confusing steps needed to run the model from GitHub. For many researchers, the incomplete setup process, errors in the existing code, and a lack of directions made it nearly impossible to run, not even considering the model's performance. We brought the code into Google Colab so users could run and evaluate problems cell-by-cell, eliminating the specific file and repository dependencies. We also provided the original training data file so users do not need to purchase the necessary computing power to train the model from the given dataset. The second change is utilizing the mutability of the code and optimizing the model so it can be used to generate images from other given inputs, such as sleep data. Additionally, the affordability of EEG technology allows for global dissemination and creates the opportunity for those who want to work on the shared DreamDiffusion model.

The automated real-time recognition of unexpected situations plays a crucial role in the safety of autonomous vehicles, especially in unsupported and unpredictable scenarios. This paper evaluates different Bayesian uncertainty quantification methods from the deep learning domain for the anticipatory testing of safety-critical misbehaviours during system-level simulation-based testing. Specifically, we compute uncertainty scores as the vehicle executes, following the intuition that high uncertainty scores are indicative of unsupported runtime conditions that can be used to distinguish safe from failure-inducing driving behaviors. In our study, we conducted an evaluation of the effectiveness and computational overhead associated with two Bayesian uncertainty quantification methods, namely MC- Dropout and Deep Ensembles, for misbehaviour avoidance. Overall, for three benchmarks from the Udacity simulator comprising both out-of-distribution and unsafe conditions introduced via mutation testing, both methods successfully detected a high number of out-of-bounds episodes providing early warnings several seconds in advance, outperforming two state-of-the-art misbehaviour prediction methods based on autoencoders and attention maps in terms of effectiveness and efficiency. Notably, Deep Ensembles detected most misbehaviours without any false alarms and did so even when employing a relatively small number of models, making them computationally feasible for real-time detection. Our findings suggest that incorporating uncertainty quantification methods is a viable approach for building fail-safe mechanisms in deep neural network-based autonomous vehicles., Comment: In Proceedings of 17th IEEE International Conference on Software Testing, Verification and Validation 2024 (ICST '24)

In scanning tunneling microscopy of molecules, an insulating buffer layer is often introduced to reduce interactions between adsorbed molecules and the substrate. Focusing on tunneling through the molecule's electronic transport gap, we demonstrate that the buffer itself strongly influences the wave function of the tunneling electron at the molecule. This is exemplified for an adsorbed platinum phthalocyanine molecule by varying the composition and thickness of the buffer. We find that, in particular, the buffer's lattice parameter is crucial. By expanding the wave function of the tunneling electron in molecular orbitals (MOs), we illustrate how one can strongly vary the relative weights of different MOs, such as the highest occupied MO versus some low-lying MOs with few nodal surfaces. The set of MOs with significant weight are important for processes used to manipulate the state of the molecule by a tunneling electron, such as molecular luminescence. The choice of buffer therefore provides an important tool for manipulating these processes., Comment: 30 pages, 5 figures

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $\rho$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $\rho$ at least as well as any state in $\mathcal{C}$ (up to some small error $\varepsilon$). This task generalizes ordinary quantum tomography of states in $\mathcal{C}$ and is more challenging because the learning algorithm must be robust to perturbations of $\rho$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $\rho$ has fidelity at least $\tau$ with a stabilizer product state, the algorithm runs in time $n^{O(1 + \log(1/\tau))} / \varepsilon^2$. This runtime is quasipolynomial in all parameters, and polynomial if $\tau$ is a constant., Comment: 20 pages. V2: minor corrections

We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $\Omega(t)$ non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $\frac{t}{2}$ bits of the true entanglement entropy., Comment: 15 pages; v2: slight edits to concurrent work section

We study real-time finite-temperature correlators for free scalars of any mass in a $dS_{d+1}$ static patch in any dimension. We show that whenever the inverse temperature is a rational multiple of the inverse de Sitter temperature, certain Matsubara poles of the symmetric Wightman function disappear. At the de Sitter temperature, we explicitly show how the Lorentzian thermal correlators can all be obtained by analytic continuations from the round $S^{d+1}$. We establish the precise relation between the Harish-Chandra character for $SO(1,d+1)$ and the integrated spectral function, providing a novel dynamical perspective on the former and enabling generalizations. Furthermore, we study scalars with exceptional non-positive masses. We provide a physical picture for the distinctive structures of their characters. For the massless case, we perform a consistent static patch quantization, and find the unique $S^{d+1}$ correlator that analytically continues to the correlators in the quantum theory., Comment: 31+10 pages, 8 figures

We define and study a variant of QMA (Quantum Merlin Arthur) in which Arthur can make multiple non-collapsing measurements to Merlin's witness state, in addition to ordinary collapsing measurements. By analogy to the class PDQP defined by Aaronson, Bouland, Fitzsimons, and Lee (2014), we call this class PDQMA. Our main result is that PDQMA = NEXP; this result builds on the PCP theorem and complements the result of Aaronson (2018) that PDQP/qpoly = ALL. While the result has little to do with quantum mechanics, we also show a more "quantum" result: namely, that QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP, under mild assumptions on the hidden-variable theory. We also observe that a quantum computer, augmented with quantum advice and the ability to inspect the history of a hidden variable, can solve any decision problem in polynomial time., Comment: 19 pages; v2: added detail to the proof of Theorem 5 and added a Main Ideas section

Deformable image registration (DIR) involves optimization of multiple conflicting objectives, however, not many existing DIR algorithms are multi-objective (MO). Further, while there has been progress in the design of deep learning algorithms for DIR, there is no work in the direction of MO DIR using deep learning. In this paper, we fill this gap by combining a recently proposed approach for MO training of neural networks with a well-known deep neural network for DIR and create a deep learning based MO DIR approach. We evaluate the proposed approach for DIR of pelvic magnetic resonance imaging (MRI) scans. We experimentally demonstrate that the proposed MO DIR approach -- providing multiple registration outputs for each patient that each correspond to a different trade-off between the objectives -- has additional desirable properties from a clinical use point-of-view as compared to providing a single DIR output. The experiments also show that the proposed MO DIR approach provides a better spread of DIR outputs across the entire trade-off front than simply training multiple neural networks with weights for each objective sampled from a grid of possible values.

Using higher-order statistics to capture cosmological information from weak lensing surveys often requires a transformation of observed shear to a measurement of the convergence signal. This inverse problem is complicated by noise and boundary effects, and various reconstruction methods have been developed to implement the process. Here we evaluate the retention of signal information of four such methods: Kaiser-Squires, Wiener filter, $\texttt{DarkMappy}$, and $\texttt{DeepMass}$. We use the higher order statistics $\textit{Minkowski functionals}$ to determine which method best reconstructs the original convergence with efficiency and precision. We find $\texttt{DeepMass}$ produces the tightest constraints on cosmological parameters, while Kaiser-Squires, Wiener filter, and $\texttt{DarkMappy}$ are similar at a smoothing scale of 3.5 arcmin. We also study the MF inaccuracy caused by inappropriate training sets in the $\texttt{DeepMass}$ method and find it to be large compared to the errors, underlining the importance of selecting appropriate training cosmologies., Comment: 14 pages, 8 figures

Using scanning tunneling microscopy (STM), we experimentally and theoretically investigate isolated platinum phthalocyanine (PtPc) molecules adsorbed on atomically thin NaCl(100) vapor deposited on Au(111). We obtain good agreement between theory and constant-height STM topography. We examine why strong distortions of STM images occur as a function of distance between molecule and STM tip. The images of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) exhibit, for increasing distance, significant radial expansion due to electron propagation in the vacuum. Additionally, the imaged angular dependence is substantially distorted. The LUMO image has substantial intensity along the molecular diagonals where PtPc has no atoms. In the electronic transport gap the image differs drastically from HOMO and LUMO, even at energies very close to these orbitals. As the tunneling becomes increasingly off-resonant, the eight angular lobes of the HOMO or of the degenerate LUMOs diminish and reveal four lobes with maxima along the molecular axes, where both, HOMO and LUMO have little or no weight. These images are strongly influenced by low-lying PtPc orbitals that have simple angular structures.

We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, $\mathsf{QEPH} = \mathsf{QRG(1)}$, the class of problems having one-turn quantum refereed games, which is known to be contained in $\mathsf{PSPACE}$. This is in contrast to the unentangled quantum polynomial hierarchy $\mathsf{QPH}$, which contains $\mathsf{QMA(2)}$. We also introduce a generalization of the quantum-classical polynomial hierarchy $\mathsf{QCPH}$ where the provers send probability distributions over strings (instead of strings) and denote it by $\mathsf{DistributionQCPH}$. Conceptually, this class is intermediate between $\mathsf{QCPH}$ and $\mathsf{QPH}$. We prove $\mathsf{DistributionQCPH} = \mathsf{QCPH}$, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that the provers can send distributions that are uniform over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., $\mathsf{DistributionPH} = \mathsf{PH}$. These results also rule out certain approaches for showing $\mathsf{QPH}$ collapses. Finally, we show that $\mathsf{PH}$ and $\mathsf{QCPH}$ are contained in $\mathsf{QPH}$, resolving an open question of Gharibian et al. (2022)., Comment: 24 pages, 1 figure, 1 table