TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions

TL;DR

TENG-BC introduces a boundary-aware neural PDE solver utilizing a natural gradient approach, enabling stable, high-precision solutions for various time-dependent PDEs with complex boundary conditions, outperforming existing methods.

Contribution

The paper presents a unified natural gradient-based framework for neural PDE solvers that effectively handles general boundary conditions and improves accuracy and stability.

Findings

Achieves solver-level accuracy on benchmark PDEs.

Handles diverse boundary conditions within a unified framework.

Outperforms conventional solvers and PINNs in experiments.

Abstract

Accurately solving time-dependent partial differential equations (PDEs) with neural networks remains challenging due to long-time error accumulation and the difficulty of enforcing general boundary conditions. We introduce TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints. At each time step, TENG-BC performs a boundary-aware optimization that jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed types within a unified framework. This formulation admits a natural-gradient interpretation, enabling stable time evolution without delicate penalty tuning. Across benchmarks over diffusion, transport, and nonlinear PDEs with various boundary conditions, TENG-BC achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.