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We introduce a generalization of the Young integration on self-similar sets defined in a closed interval and give a sufficient condition of its integrability. We also prove integration by substitution, integration by parts and term-by-term integration and give examples of the properties., Comment: 24 pages

The ability to perform fast and accurate atomistic simulations is crucial for advancing the chemical sciences. By learning from high-quality data, machine-learned interatomic potentials achieve accuracy on par with ab initio and first-principles methods at a fraction of their computational cost. The success of machine-learned interatomic potentials arises from integrating inductive biases such as equivariance to group actions on an atomic system, e.g., equivariance to rotations and reflections. In particular, the field has notably advanced with the emergence of equivariant message passing. Most of these models represent an atomic system using spherical tensors, tensor products of which require complicated numerical coefficients and can be computationally demanding. Cartesian tensors offer a promising alternative, though state-of-the-art methods lack flexibility in message-passing mechanisms, restricting their architectures and expressive power. This work explores higher-rank irreducible Cartesian tensors to address these limitations. We integrate irreducible Cartesian tensor products into message-passing neural networks and prove the equivariance and traceless property of the resulting layers. Through empirical evaluations on various benchmark data sets, we consistently observe on-par or better performance than that of state-of-the-art spherical and Cartesian models., Comment: Accepted at NeurIPS 2024 (camera-ready version)

Inverse design, where we seek to design input variables in order to optimize an underlying objective function, is an important problem that arises across fields such as mechanical engineering to aerospace engineering. Inverse design is typically formulated as an optimization problem, with recent works leveraging optimization across learned dynamics models. However, as models are optimized they tend to fall into adversarial modes, preventing effective sampling. We illustrate that by instead optimizing over the learned energy function captured by the diffusion model, we can avoid such adversarial examples and significantly improve design performance. We further illustrate how such a design system is compositional, enabling us to combine multiple different diffusion models representing subcomponents of our desired system to design systems with every specified component. In an N-body interaction task and a challenging 2D multi-airfoil design task, we demonstrate that by composing the learned diffusion model at test time, our method allows us to design initial states and boundary shapes that are more complex than those in the training data. Our method generalizes to more objects for N-body dataset and discovers formation flying to minimize drag in the multi-airfoil design task. Project website and code can be found at https://github.com/AI4Science-WestlakeU/cindm., Comment: ICLR 2024 spotlight. 30 pages, 17 figures

Long-range interactions are essential for the correct description of complex systems in many scientific fields. The price to pay for including them in the calculations, however, is a dramatic increase in the overall computational costs. Recently, deep graph networks have been employed as efficient, data-driven surrogate models for predicting properties of complex systems represented as graphs. These models rely on a local and iterative message passing strategy that should, in principle, capture long-range information without explicitly modeling the corresponding interactions. In practice, most deep graph networks cannot really model long-range dependencies due to the intrinsic limitations of (synchronous) message passing, namely oversmoothing, oversquashing, and underreaching. This work proposes a general framework that learns to mitigate these limitations: within a variational inference framework, we endow message passing architectures with the ability to freely adapt their depth and filter messages along the way. With theoretical and empirical arguments, we show that this simple strategy better captures long-range interactions, by surpassing the state of the art on five node and graph prediction datasets suited for this problem. Our approach consistently improves the performances of the baselines tested on these tasks. We complement the exposition with qualitative analyses and ablations to get a deeper understanding of the framework's inner workings.

Simulating the time evolution of physical systems is pivotal in many scientific and engineering problems. An open challenge in simulating such systems is their multi-resolution dynamics: a small fraction of the system is extremely dynamic, and requires very fine-grained resolution, while a majority of the system is changing slowly and can be modeled by coarser spatial scales. Typical learning-based surrogate models use a uniform spatial scale, which needs to resolve to the finest required scale and can waste a huge compute to achieve required accuracy. In this work, we introduce Learning controllable Adaptive simulation for Multi-resolution Physics (LAMP) as the first full deep learning-based surrogate model that jointly learns the evolution model and optimizes appropriate spatial resolutions that devote more compute to the highly dynamic regions. LAMP consists of a Graph Neural Network (GNN) for learning the forward evolution, and a GNN-based actor-critic for learning the policy of spatial refinement and coarsening. We introduce learning techniques that optimizes LAMP with weighted sum of error and computational cost as objective, allowing LAMP to adapt to varying relative importance of error vs. computation tradeoff at inference time. We evaluate our method in a 1D benchmark of nonlinear PDEs and a challenging 2D mesh-based simulation. We demonstrate that our LAMP outperforms state-of-the-art deep learning surrogate models, and can adaptively trade-off computation to improve long-term prediction error: it achieves an average of 33.7% error reduction for 1D nonlinear PDEs, and outperforms MeshGraphNets + classical Adaptive Mesh Refinement (AMR) in 2D mesh-based simulations. Project website with data and code can be found at: http://snap.stanford.edu/lamp., Comment: ICLR 2023, notable top-25% (spotlight), 19 pages, 9 figures

In this paper, we generalize the Cantor function to $2$-dimensional cubes and construct a cyclic $2$-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the $2$-dimensional torus with the generalized Cantor function while it vanishes on the Lipschitz functions on the Cantor dust. The cocycle is calculated through the integration of $2$-forms on the torus by using a combinatorial Fredholm module., Comment: 8 pages

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. LE-PDE learns a compact, global representation of the system and efficiently evolves it fully in the latent space with learned latent evolution models. LE-PDE achieves speed-up by having a much smaller latent dimension to update during long rollout as compared to updating in the input space. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We further introduce techniques for speeding-up inverse optimization of boundary conditions for PDEs via backpropagation through time in latent space, and an annealing technique to address the non-differentiability and sparse interaction of boundary conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D Navier-Stokes flows into turbulent phase and an inverse optimization of boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep learning-based surrogate models and other strong baselines, we demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy., Comment: NeurIPS 2022; 30 pages, 15 figures

The Heavy Photon Search (HPS) experiment is designed to search for a new vector boson $A^\prime$ in the mass range of 20 MeV/$c^2$ to 220 MeV/$c^2$ that kinetically mixes with the Standard Model photon with couplings $\epsilon^2 >10^{-10}$. In addition to the general importance of exploring light, weakly coupled physics that is difficult to probe with high-energy colliders, a prime motivation for this search is the possibility that sub-GeV thermal relics constitute dark matter, a scenario that requires a new comparably light mediator, where models with a hidden $U(1)$ gauge symmetry, a "dark", "hidden sector", or "heavy" photon, are particularly attractive. HPS searches for visible signatures of these heavy photons, taking advantage of their small coupling to electric charge to produce them via a process analogous to bremsstrahlung in a fixed target and detect their subsequent decay to $\mathrm{e}^+ \mathrm{e}^-$ pairs in a compact spectrometer. In addition to searching for $\mathrm{e}^+ \mathrm{e}^-$ resonances atop large QED backgrounds, HPS has the ability to precisely measure decay lengths, resulting in unique sensitivity to dark photons, as well as other long-lived new physics. After completion of the experiment and operation of engineering runs in 2015 and 2016 at the JLab CEBAF, physics runs in 2019 and 2021 have provided datasets that are now being analyzed to search for dark photons and other new phenomena., Comment: Submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module., Comment: 31 pages, to appear in J. Fractal Geom

We present an initial design study for LDMX, the Light Dark Matter Experiment, a small-scale accelerator experiment having broad sensitivity to both direct dark matter and mediator particle production in the sub-GeV mass region. LDMX employs missing momentum and energy techniques in multi-GeV electro-nuclear fixed-target collisions to explore couplings to electrons in uncharted regions that extend down to and below levels that are motivated by direct thermal freeze-out mechanisms. LDMX would also be sensitive to a wide range of visibly and invisibly decaying dark sector particles, thereby addressing many of the science drivers highlighted in the 2017 US Cosmic Visions New Ideas in Dark Matter Community Report. LDMX would achieve the required sensitivity by leveraging existing and developing detector technologies from the CMS, HPS and Mu2e experiments. In this paper, we present our initial design concept, detailed GEANT-based studies of detector performance, signal and background processes, and a preliminary analysis approach. We demonstrate how a first phase of LDMX could expand sensitivity to a variety of light dark matter, mediator, and millicharge particles by several orders of magnitude in coupling over the broad sub-GeV mass range.

We define a variant of the Young integration on some kinds of self-similar sets which are called cellular self-similar sets. This variant is an analogue of the Young integration defined on the unit interval. We give the criteria of the variant on cellular self-similar sets, and also show that the variant is a cyclic 1-cocycle of the algebra of complex-valued H\"older continuous functions on the cellular self-similar sets. This suggests that the cocycle is a variant of currents., Comment: 26 pages

This report, based on the Dark Sectors workshop at SLAC in April 2016, summarizes the scientific importance of searches for dark sector dark matter and forces at masses beneath the weak-scale, the status of this broad international field, the important milestones motivating future exploration, and promising experimental opportunities to reach these milestones over the next 5-10 years., Comment: 66 pages, 15 figures, 3 tables. Workshop website and agenda: http://www-conf.slac.stanford.edu/darksectors2016/ https://indico.cern.ch/event/507783/ Editors: J. Alexander, M. Battaglieri, B. Echenard, R. Essig, M. Graham, E. Izaguirre, J. Jaros, G. Krnjaic, J. Mardon, D. Morrissey, T. Nelson, M. Perelstein, M. Pyle, A. Ritz, P. Schuster, B. Shuve, N. Toro, R. Van De Water

The Heavy Photon Search (HPS), an experiment to search for a hidden sector photon in fixed target electroproduction, is preparing for installation at the Thomas Jefferson National Accelerator Facility (JLab) in the Fall of 2014. As the first stage of this project, the HPS Test Run apparatus was constructed and operated in 2012 to demonstrate the experiment's technical feasibility and to confirm that the trigger rates and occupancies are as expected. This paper describes the HPS Test Run apparatus and readout electronics and its performance. In this setting, a heavy photon can be identified as a narrow peak in the e$^+$e$^-$ invariant mass spectrum, above the trident background or as a narrow invariant mass peak with a decay vertex displaced from the production target, so charged particle tracking and vertexing are needed for its detection. In the HPS Test Run, charged particles are measured with a compact forward silicon microstrip tracker inside a dipole magnet. Electromagnetic showers are detected in a PbW0$_{4}$ crystal calorimeter situated behind the magnet, and are used to trigger the experiment and identify electrons and positrons. Both detectors are placed close to the beam line and split top-bottom. This arrangement provides sensitivity to low-mass heavy photons, allows clear passage of the unscattered beam, and avoids the spray of degraded electrons coming from the target. The discrimination between prompt and displaced e$^+$e$^-$ pairs requires the first layer of silicon sensors be placed only 10~cm downstream of the target. The expected signal is small, and the trident background huge, so the experiment requires very large statistics. Accordingly, the HPS Test Run utilizes high-rate readout and data acquisition electronics and a fast trigger to exploit the essentially 100% duty cycle of the CEBAF accelerator at JLab., Comment: Revised version to match published version, 16 pages, 18 figures, published in Nuclear Instruments and Methods in Physics Research Section A, editor: Per Hansson Adrian

Future linear colliders will require high levels of performance from their electron sources. A group at SLAC has recently tested a structure that substantially exceeds current collider polarized electron source pulse-profile requirements., Comment: 6 pages, 4 figures