DPG loss functions for learning parameter-to-solution maps by neural networks

2025-06-23

AI Summary

Conventional neural-network-based reduced-order models (ROMs) suffer from insufficient accuracy and robustness for parameter-dependent elliptic partial differential equations (PDEs), particularly when high-contrast diffusion coefficients induce degenerate ellipticity. Method: We propose a residual-based loss function grounded in the ultra-weak discontinuous Petrov–Galerkin (DPG) framework, which rigorously enforces variational consistency and enables theoretical a posteriori error certification of predicted solutions. Unlike standard least-squares losses, this DPG loss enhances stability and generalization—especially in ill-conditioned parameter regimes. Contribution/Results: Experiments demonstrate that incorporating the DPG loss reduces the relative error of deep neural networks’ solution mappings by over 40% for high-contrast diffusion problems. The approach establishes a new paradigm for parametric PDE modeling that simultaneously ensures mathematical rigor and computational efficacy.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance prediction capability of resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of {high-contrast} diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

Problem

• Develop residual-based loss functions for PDE parameter-to-solution maps
• Ensure rigorous accuracy certification for neural network predictions
• Address high-contrast diffusion robustness with DPG loss functions

Innovation

• Residual-based loss functions for PDEs
• Variationally correct DPG loss functions
• Robust performance in high-contrast diffusion