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A statistical mechanics framework for constructing non-equilibrium thermodynamic models

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arXiv 2023年

作者： Leadbetter, Travis Purohit, Prashant K. Reina, Celia Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of non-equilibrium systems. But despite tremendous efforts, a universal theory of non-equilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex non-equilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data, or on running concurrent simulations at the particle level. Instead, we provide a framework for deriving stand-alone macroscopic thermodynamics models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework's effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double stranded DNA, thus producing, to the authors' knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers. © 2023, CC BY.

关键词： Statistical mechanics

LEARNING ONLY ON BOUNDARIES: A PHYSICS-INFORMED NEURAL OPERATOR FOR SOLVING PARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS IN COMPLEX GEOMETRIES

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arXiv 2023年

作者： Fang, Zhiwei Wang, Sifan Perdikaris, Paris Graduate Group in Applied Mathematics Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanichal Engineering Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from O(Nd) to O(Nd−1), where d is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Boundary element method

An Expert's Guide to Training Physics-informed Neural Networks

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arXiv 2023年

作者： Wang, Sifan Wang, Hanwen Sankaran, Shyam Perdikaris, Paris Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

Physics-informed neural networks (PINNs) have been popularized as a deep learning framework that can seamlessly synthesize observational data and partial differential equation (PDE) constraints. Their practical effectiveness however can be hampered by training pathologies, but also oftentimes by poor choices made by users who lack deep learning expertise. In this paper we present a series of best practices that can significantly improve the training efficiency and overall accuracy of PINNs. We also put forth a series of challenging benchmark problems that highlight some of the most prominent difficulties in training PINNs, and present comprehensive and fully reproducible ablation studies that demonstrate how different architecture choices and training strategies affect the test accuracy of the resulting models. We show that the methods and guiding principles put forth in this study lead to state-of-the-art results and provide strong baselines that future studies should use for comparison purposes. To this end, we also release a highly optimized library in JAX that can be used to reproduce all results reported in this paper, enable future research studies, as well as facilitate easy adaptation to new use-case scenarios. © 2023, CC BY-NC-SA.

关键词： Deep learning

Moving from Machine Learning to Statistics: the case of Expected Points in American football

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arXiv 2024年

作者： Brill, Ryan S. Yee, Ryan Deshpande, Sameer K. Wyner, Abraham J. Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania United States Dept. of Statistics University of Wisconsin–Madison United States Dept. of Statistics and Data Science The Wharton School University of Pennsylvania United States

Expected points is a value function fundamental to player evaluation and strategic in-game decision-making across sports analytics, particularly in American football. To estimate expected points, football analysts use machine learning tools, which are not equipped to handle certain challenges. They suffer from selection bias, display counter-intuitive artifacts of overfitting, do not quantify uncertainty in point estimates, and do not account for the strong dependence structure of observational football data. These issues are not unique to American football or even sports analytics;they are general problems analysts encounter across various statistical applications, particularly when using machine learning in lieu of traditional statistical models. We explore these issues in detail and devise expected points models that account for them. We also introduce a widely applicable novel methodological approach to mitigate overfitting, using a catalytic prior to smooth our machine learning models. © 2024, CC BY.

关键词： Adversarial machine learning

ENSEMBLE LEARNING FOR PHYSICS INFORMED NEURAL NETWORKS: A GRADIENT BOOSTING APPROACH

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arXiv 2023年

作者： Fang, Zhiwei Wang, Sifan Perdikaris, Paris Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mechanichal Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date, conventional PINNs have not been successful in simulating multi-scale and singular perturbation problems. In this work, we present a new training paradigm referred to as "gradient boosting" (GB), which significantly enhances the performance of physics informed neural networks (PINNs). Rather than learning the solution of a given partial differential equation (PDE) using a single neural network directly, our algorithm employs a sequence of neural networks to achieve a superior outcome. This approach allows us to solve problems presenting great challenges for traditional PINNs. Our numerical experiments demonstrate the effectiveness of our algorithm through various benchmarks, including comparisons with finite element methods and PINNs. Furthermore, this work also unlocks the door to employing ensemble learning techniques in PINNs, providing opportunities for further improvement in solving PDEs. © 2023, CC BY-NC-SA.

关键词： Numerical methods

Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes

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arXiv 2023年

作者： Jasra, Ajay Kamatani, Kengo Maama, Mohamed Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Institute of Statistical Mathematics Tokyo190-0014 Japan

In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes (PDMPs). In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in [20], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameter of that algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples. Copyright © 2023, The Authors. All rights reserved.

关键词： Ordinary differential equations

Unbiased Parameter Estimation for Bayesian Inverse Problems

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arXiv 2025年

作者： Chada, Neil K. Jasra, Ajay Maama, Mohamed Tempone, Raul Department of Mathematics City University of Hong Kong China School of Data Science The Chinese University of Hong Kong Shenzhen CN Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximizer of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

SR-CLD: spatially-resolved chord length distributions for statistical description, visualization, and alignment of non-uniform microstructures

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arXiv 2024年

作者： Whitman, Sheila E. Latypov, Marat I. Graduate Interdisciplinary Program in Applied Mathematics University of Arizona TucsonAZ85721 United States Department of Materials Science and Engineering University of Arizona TucsonAZ85721 United States

This study introduces the calculation of spatially-resolved chord length distribution (SR-CLD) as an efficient approach for quantifying and visualizing non-uniform microstructures in heterogeneous materials. SR-CLD enables detailed analysis of spatial variation of microstructure constituent sizes in different directions that can be overlooked with traditional descriptions. We present the calculation of SR-CLD using efficient scan-line algorithm that counts pixels in constituents along pixel rows or columns of microstructure images for detailed, high-resolution SR-CLD maps. We demonstrate the application of SR-CLD in two case studies: one on synthetic polycrystalline microstructures with known and intentionally created uniform and gradient spatial distributions of grain size;and one on experimental images of two-phase microstructures of additively manufactured Ti alloys with significant spatially non-uniform distributions of laths of one of the phases. Additionally, we present how SR-CLDs can enable automated, computationally efficient, and robust alignment of large sets of images for merging into accurate composite images of large microstructure areas. © 2024, CC BY.

关键词： Titanium alloys

AnisoGNN: graph neural networks generalizing to anisotropic properties of polycrystals

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arXiv 2024年

作者： Hu, Guangyu Latypov, Marat I. Department of Materials Science and Engineering University of Arizona TucsonAZ85721 United States Graduate Interdisciplinary Program in Applied Mathematics University of Arizona TucsonAZ85721 United States

We present AnisoGNNs - graph neural networks (GNNs) that generalize predictions of anisotropic properties of polycrystals in arbitrary testing directions without the need in excessive training data. To this end, we develop GNNs with a physics-inspired combination of node attributes and aggregation function. We demonstrate the excellent generalization capabilities of AnisoGNNs in predicting anisotropic elastic and inelastic properties of two alloys. © 2024, CC BY.

关键词： Anisotropy