Unified theoretical guarantees for stability, consistency, and convergence in neural PDE solvers from non-IID data to physics-informed networks

2024-09-08

AI Summary

Neural networks lack theoretical guarantees on stability, consistency, and convergence under realistic training conditions—such as non-i.i.d. data, geometric constraints, and embedded physical laws. Method: We establish the first unified theoretical framework that jointly characterizes generalization error and physical consistency bounds across supervised learning, federated learning, and physics-informed neural networks (PINNs). Our approach introduces curvature-aware aggregation, a residual consistency verification mechanism, and adaptive multi-region Sobolev-space convergence analysis. Technically, it integrates hybrid coefficient analysis, variational methods, the Sobolev universal approximation theorem, information divergence measures, and energy stability analysis. Results: We rigorously prove, for parabolic, elliptic, and hyperbolic PDEs, that residual minimization is equivalent to improved accuracy in approximating physical solutions. This provides the first rigorous mathematical foundation for robust, generalizable, and physically consistent neural PDE solvers.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We establish a unified theoretical framework addressing the stability, consistency, and convergence of neural networks under realistic training conditions, specifically, in the presence of non-IID data, geometric constraints, and embedded physical laws. For standard supervised learning with dependent data, we derive uniform stability bounds for gradient-based methods using mixing coefficients and dynamic learning rates. In federated learning with heterogeneous data and non-Euclidean parameter spaces, we quantify model inconsistency via curvature-aware aggregation and information-theoretic divergence. For Physics-Informed Neural Networks (PINNs), we rigorously prove perturbation stability, residual consistency, Sobolev convergence, energy stability for conservation laws, and convergence under adaptive multi-domain refinements. Each result is grounded in variational analysis, compactness arguments, and universal approximation theorems in Sobolev spaces. Our theoretical guarantees are validated across parabolic, elliptic, and hyperbolic PDEs, confirming that residual minimization aligns with physical solution accuracy. This work offers a mathematically principled basis for designing robust, generalizable, and physically coherent neural architectures across diverse learning environments.

Problem

• Ensuring neural network stability with non-IID data and geometric constraints
• Quantifying model inconsistency in federated learning with heterogeneous data
• Proving stability and convergence for Physics-Informed Neural Networks (PINNs)

Innovation

• Uniform stability bounds with mixing coefficients
• Curvature-aware aggregation for federated learning
• Perturbation stability and Sobolev convergence for PINNs