Data-driven learning of robust nonlocal physics from high-fidelity synthetic data

Abstract

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting provably robust data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity. Our scheme allows extraction of provably invertible nonlocal models whose kernels may be partially negative. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints and ensure that models are robust, even in small-data regimes. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.

32 pages, 10 figures, 3 tables

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• Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks
• Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows
• Tempered Fractional LES Modeling
• An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
• A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems
• Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations
• Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
• Efficient quadrature rules for finite element discretizations of nonlocal equations
• ML-based identification of the interface regions for coupling local and nonlocal models
• Efficient optimization-based quadrature for variational discretization of nonlocal problems