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

2026-02-27

AI Summary

This work addresses the challenges of long-term error accumulation and inaccurate enforcement of boundary conditions that commonly plague existing neural network approaches for solving time-dependent partial differential equations with general boundary conditions. The authors propose TENG-BC, a novel method that, for the first time, supports Dirichlet, Neumann, Robin, and mixed boundary conditions within a unified framework. By incorporating boundary-aware optimization, TENG-BC jointly handles interior dynamics and boundary constraints at each time step and introduces a time-evolution natural gradient to circumvent the need for tedious hyperparameter tuning of penalty terms typical in physics-informed neural networks (PINNs). Experiments demonstrate that TENG-BC achieves accuracy comparable to conventional numerical solvers on benchmark problems—including diffusion, transport, and nonlinear PDEs—with similar sampling costs, significantly outperforming existing PINN-based methods.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
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• Machine Learning: Learning with Manifolds

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• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
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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.

Problem

• neural PDE solvers
• time-dependent PDEs
• general boundary conditions
• error accumulation
• boundary enforcement

Innovation

• Time-Evolving Natural Gradient
• Neural PDE Solver
• General Boundary Conditions
• Boundary-Aware Optimization
• Physics-Informed Neural Networks