A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs
We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first introduce the "extra fields" from the mixed finite element method to reformulate the PDEs so as to equivalently transform the three types of BCs into linear equations. Based on the reformulation, we derive the general solutions of the BCs analytically, which are employed to construct an ansatz that automatically satisfies the BCs. With such a framework, we can train the neural networks without adding extra loss terms and thus efficiently handle geometrically complex PDEs, alleviating the unbalanced competition between the loss terms corresponding to the BCs and PDEs. We theoretically demonstrate that the "extra fields" can stabilize the training process. Experimental results on real-world geometrically complex PDEs showcase the effectiveness of our method compared with state-of-the-art baselines.
Code (1)
Similar Papers 제목 키워드 기반
Trusted Neural Networks for Safety-Constrained Autonomous Control
We propose Trusted Neural Network (TNN) models, which are deep neural network models that satisfy safety constraints critical to the application domain. We investigate different mechanisms for incorporating rule-based kn…
Self-Driving CarsFace Relighting with Geometrically Consistent Shadows
Most face relighting methods are able to handle diffuse shadows, but struggle to handle hard shadows, such as those cast by the nose. Methods that propose techniques for handling hard shadows often do not produce geometr…
A Unified Knowledge Embedded Reinforcement Learning-based Framework for Generalized Capacitated Vehicle Routing Problems
The Capacitated Vehicle Routing Problem (CVRP) is a fundamental NP-hard problem with broad applications in logistics and transportation. Real-world CVRPs often involve diverse objectives and complex constraints, such as …
Reinforcement LearningSolving DCOPs with Distributed Large Neighborhood Search
The field of Distributed Constraint Optimization has gained momentum in recent years, thanks to its ability to address various applications related to multi-agent cooperation. Nevertheless, solving Distributed Constraint…
Neuro-Symbolic Verification on Instruction Following of LLMs
A fundamental problem of applying Large Language Models (LLMs) to important applications is that LLMs do not always follow instructions, and violations are often hard to observe or check. In LLM-based agentic workflows, …
Instruction FollowingLogical Reasoning