The theory of homogenization provides a systematic approach to the derivation macroscale constitutive laws, obviating need repeatedly resolve complex microstructure. However, unit cell problem that defines model is typically not amenable explicit evaluation. It therefore interest learn models from data generated by problem. Many viscoelastic and elastoviscoplastic materials are characterized memory-dependent laws. In order amortize computational investment in finding such it desirable their...

Neural dynamical systems are that described at least in part by neural networks. The class of continuous-time must, however, be numerically integrated for simulation and learning. Here, we present a compact circuit two common numerical integrators: the explicit fixed-step Runge-Kutta method any order semi-implicit/predictor-corrector Adams-Bashforth-Moulton method. Modeled as constant-sized recurrent networks embedding continuous differential equation, they achieve fully temporal output....

Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven that map between function spaces. However, instead of full-field measurements, often the available data are only finite-dimensional parametrizations model inputs or finite observables outputs. Building off Fourier Neural Operators, this paper introduces Mappings (FNMs) framework is able to accommodate such The develops universal...

Operator learning is a variant of machine that designed to approximate maps between function spaces from data. The Fourier Neural (FNO) common model architecture used for operator learning. FNO combines pointwise linear and nonlinear operations in physical space with space, leading parameterized map acting spaces. Although FNOs formally involve convolutions functions on continuum, practice the computations are performed discretized grid, allowing efficient implementation via FFT. In this...

In stochastic systems, informative approaches select key measurement or decision variables that maximize information gain to enhance the efficacy of model-related inferences. Neural Learning also embodies dynamics, but is less developed. Here, we propose Informative Ensemble Kalman Learning, which replaces backpropagation with an adaptive Filter quantify uncertainty and enables maximizing during Learning. After demonstrating Learning's competitive performance on standard datasets, apply...

The optimal design of neural networks is a critical problem in many applications. Here, we investigate how dynamical systems with polynomial nonlinearities can inform the that seek to emulate them. We propose Learnability metric and its associated features quantify near-equilibrium behavior learning dynamics. Equating equivalent parameter estimation reference system establishes bounds on network structure. In this way, norms from theory provide good first guess for structure, which may then...

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting simplified are computationally tractable while accurately predicting macroscopic response. In field of continuum mechanics, homogenization crucial for deriving constitutive laws incorporate microscale physics order formulate balance quantities...

Fully resolving dynamics of materials with rapidly-varying features involves expensive fine-scale computations which need to be conducted on macroscopic scales. The theory homogenization provides an approach derive effective equations eliminates the small scales by exploiting scale separation. An accurate homogenized model avoids computationally-expensive task numerically solving underlying balance laws at a fine scale, thereby rendering numerical solution more computationally tractable. In...

This paper concerns the study of history dependent phenomena in heterogeneous materials a two-scale setting where material is specified at fine microscopic scale heterogeneities that much smaller than coarse macroscopic application. We specifically polycrystalline medium each grain governed by crystal plasticity while solid subjected to dynamic loads. The theory homogenization allows us solve macroscale problem directly with constitutive relation defined implicitly solution microscale...