The Inference Report

We study the learning of homogenized constitutive laws for inelastic heterogeneous materials, aiming to replace costly concurrent unit-cell simulations with surrogates that generalize across loading histories and microstructures. We focus on the recurrent neural operator framework, motivated by the explicit form of homogenized Kelvin--Voigt linear viscoelasticity. In this framework, internal variables enable a Markovian approximation of loading-history dependence, and neural operators represent the microstructure-dependent constitutive response. For periodic Kelvin--Voigt viscoelasticity, we prove a universal approximation theorem showing that the model can approximate the homogenized constitutive law to any prescribed accuracy, uniformly in time and over a wide set of admissible strain trajectories and microstructural property fields, in two and three dimensions. This comprises a significant extension of the existing, somewhat limited, one-dimensional theory that uses explicit solutions for piecewise constant materials. Instead, in two and three dimensions, we adopt a more versatile approximation based on the spectral representation of the memory kernel that arises from homogenization. This also enables us to give an explicit bound on the internal-variable dimension, sufficient for approximation, exhibiting logarithmic growth in the inverse relative stress error and in material contrast. Numerical experiments on viscoelasticity and crystal plasticity complement the theory, illustrating the roles of internal-variable dimension and sample complexity in learning homogenized responses from unit-cell data. Theoretical results are formally verified in Lean 4.

Training Gaussian processes (GPs) at large scale is computationally challenging because evaluating the log-marginal likelihood and its gradients involves large, dense kernel matrices. We develop a scalable low-rank GP training approach that combines AnchorNet landmark selection with a fully independent training conditional (FITC) correction. The resulting objective and its analytic gradients can be evaluated using low-rank factors, diagonal quantities, and small dense matrices, without forming the full kernel matrix or its derivatives. The complete training procedure is implemented in single precision and evaluated on nine large-scale regression datasets. Across these problems, the method reaches the reference predictive accuracy while substantially reducing training time, although the required approximation rank varies across datasets. We also study the numerical behavior of the full-kernel predictive solve and show that single precision can become limiting for ill-conditioned problems. Finally, we investigate further reducing selected matrix operations to fp16 or TF32. These lower-precision variants provide only modest additional speedups and can become unreliable when the reduced system is poorly conditioned.

Kinetic equations in the high-field regime describe particle transport dominated by electric-field acceleration and collisional relaxation. These equations are widely used in simulations of plasmas and semiconductor devices. Typical models include the Vlasov--Poisson--Fokker--Planck (VPFP) and Boltzmann--Poisson systems. Their high dimensionality and multiscale nature pose two of the major challenges to accurate and efficient numerical solution. To address these challenges, we propose an asymptotic-preserving tensor neural network (APTNN) method. Specifically, to reduce high-dimensional computational costs and memory requirements, we adopt the canonical polyadic (CP) decomposition structure of TNNs to approximate their solutions. We show that the equation residuals retain this decomposition form, allowing high-dimensional loss integrals to be evaluated through combinations of one-dimensional integrals. The resulting computational and storage costs are reduced from exponential to polynomial growth with dimension. To improve the prediction accuracy in the high-field limit, we enhance the baseline AP loss by incorporating the first moment equation. The added loss term directly constrains the limiting flux closure to reduce its approximation error in the high-field regime. The enhanced AP loss can still be evaluated dimension-wise. Numerical results show that APTNN improves prediction accuracy in the high-field regime with significantly reduced computational costs and memory usage. For a 3D3V VPFP problem with multiple random inputs, training takes less than 30 minutes on a single GPU.

Model reduction via the Mori-Zwanzig (MZ) formalism is well understood for coarse-grained representations constructed from standard basis functions such as Fourier functions. We show that the same formalism can be used to build reduced-order models in bases discovered by physics-informed machine learning. The fluctuation-dissipation theorem (FDT) is used to close the resulting non-Markovian evolution equations for the expansion coefficients of the learned basis. We further introduce a complementary, generative closure model. Relative to the Markovian dynamics alone, both the MZ and generative closures substantially improve predictions of the filtered states.

The perfect electric conductor (PEC) boundary condition alone does not eliminate the gradient nullspace of the Maxwell curl-curl operator. Consequently, a boundary-only Maxwell implementation of Eig-PIELM, an eigenvalue extension of the physics-informed extreme learning machine (PIELM), may retain represented gradient modes in its reduced trial space, producing nonphysical, gradient-contaminated modes. We develop dR-Eig-PIELM, a finite-dimensional structure-preserving extension that retains the linear-coefficient, boundary-null-space, and one-shot multimode architecture of Eig-PIELM. It combines compatible tensor-product grad--curl trial spaces with a constrained null-space construction to restrict the sampled PEC space to the mass-orthogonal complement of the represented gradient image before the eigensolve. Symmetric weak-form Gram assembly yields a symmetric-definite reduced eigenproblem without an eigenvalue cutoff under the stated sampling and material-positivity assumptions. For rectangles and cuboids, we establish fixed-degree exactness at the PEC vector-field space and derive rank and dimension identities. Numerical experiments show that dR-Eig-PIELM excludes the spurious eigenmodes with significant gradient components observed in the boundary-only baseline. The method exhibits rapid convergence with increasing polynomial degree and competitive work--precision performance in the tested configurations. A covariantly mapped waveguide example illustrates the potential of dR-Eig-PIELM for integration into design optimization workflows through a constrained local geometry update driven by computed cutoff frequencies and their sensitivities. This work provides a mathematical and computational framework for separating represented gradient modes from physical Maxwell eigenmodes within Eig-PIELM and supports its use in waveguide spectral analysis and geometry design.

Reduced-order models (ROMs) learned from trusted numerical simulations enable the real-time simulation of complex physical systems, making computationally prohibitive tasks-such as uncertainty quantification and optimal control-tractable. In particular, data-driven ROMs are opening the opportunity to learn from both numerical simulations and experimental data. However, integrating these sources of information is far from straightforward. Real-world measurements often involve multiple sources of uncertainty, including non-Gaussian noise, latent heterogeneities, unknown or hidden system parameters, and multiscale phenomena occurring below the resolution of the data. Consequently, real-world data are better described by parametrized stochastic partial differential equations, for which the development of effective ROMs remains relatively unexplored. In this work, we introduce POD-CNF, a data-driven generative ROM for efficient uncertainty quantification of parametrized high-dimensional systems. The approach combines projection-based model reduction and deep generative modeling. First, Proper Orthogonal Decomposition (POD) is employed to obtain a low-dimensional representation of the solution manifold. A Conditional Normalizing Flow (CNF) is then trained to approximate the density of the reduced coefficients conditionally on the input parameters. This formulation allows the surrogate to capture complex, multimodal output distributions, going beyond standard regression-based ROMs and least-squares predictions. We assess the effectiveness of POD-CNF through numerical experiments, demonstrating its ability to accurately reproduce output distributions while substantially reducing computational costs. Finally, a proof-of-concept application illustrates how the generative paradigm facilitates integration into Bayesian inverse problems, highlighting promising directions for future research.