Unified theoretical guarantees for stability, consistency, and convergence in neural PDE solvers from non-IID data to physics-informed networks
This provides a mathematically principled foundation for designing robust neural PDE solvers, addressing critical issues in scientific machine learning.
The authors developed a unified theoretical framework to analyze the stability, consistency, and convergence of neural networks in solving partial differential equations (PDEs) under realistic conditions like non-IID data and physical constraints, proving rigorous guarantees for various learning scenarios including PINNs.
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.