Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates

TL;DR

This paper introduces VMLFN, a physics-informed neural network that efficiently creates multiphysics surrogates by reformulating PDEs into a linear matrix problem, enabling fast and accurate predictions.

Contribution

The paper presents a novel variational matrix-learning Fourier network that simplifies physics-informed training into a linear problem, avoiding complex differentiation and tuning.

Findings

VMLFN achieves high accuracy in heat conduction, solid mechanics, and wave problems.

It provides substantial speedup over traditional finite-element and neural network methods.

The method effectively captures the spectral range of target problems with a heuristic frequency scan.

Abstract

Multiphysics simulation is critical for system-technology co-optimization (STCO) in chiplet-based design, but repeated finite-element solutions of PDE-governed problems are computationally expensive in parametric design exploration. This paper proposes a variational matrix-learning Fourier network (VMLFN) for efficient parametric multiphysics surrogate modeling. VMLFN constructs a log-space sine neural representation with randomly sampled spectral frequencies, frequency-dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden-layer parameters, the output-layer weights are determined by reformulating the governing PDEs into variational weak forms and enforcing the stationarity condition of the resulting energy functional. This converts physics-informed training into a linear matrix-solving problem, requiring only first-order derivatives and avoiding both high-order automatic differentiation and penalty-coefficient tuning. A heuristic frequency-scanning algorithm is further introduced to select a problem-adaptive maximum frequency that covers the dominant spectral range of the target problem. The proposed method is validated on heat conduction, solid mechanics, and Helmholtz wave propagation problems. Results from five benchmark cases demonstrate that VMLFN delivers accurate full-field predictions with substantial speedup over conventional physics-informed neural networks and repeated finite-element simulations.