Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates

2026-05-04

AI Summary

This work addresses the high computational cost of multiphysics parametric simulation in chiplet design, which arises from repeatedly solving partial differential equations. To overcome this challenge, the authors propose the Variational Matrix Learning Fourier Network (VMLFN), a novel approach that reformulates governing equations into their variational weak forms and leverages the stationarity condition of an energy functional to cast physics-informed training as a linear matrix problem. The method innovatively integrates logarithmic-space sinusoidal neural representations, frequency-adaptive decay modulation, embedded Dirichlet boundary conditions, and a heuristic frequency-scanning algorithm to automatically identify dominant spectral ranges. By circumventing high-order automatic differentiation and eliminating the need for penalty coefficient tuning, VMLFN achieves highly accurate full-field predictions across five benchmark cases—including heat conduction, solid mechanics, and Helmholtz wave propagation—significantly outperforming conventional physics-informed neural networks and repeated finite element simulations.

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.

Problem

• multiphysics simulation
• parametric design
• computational efficiency
• PDE-governed problems
• system-technology co-optimization

Innovation

• variational formulation
• Fourier neural networks
• physics-informed machine learning
• matrix-learning
• parametric surrogate modeling