Hard constraint learning approaches with trainable influence functions for evolutionary equations
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Abstract (arXiv)
This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low computational accuracy of standard Physics-Informed Neural Networks (PINNs) in large temporal domains.Sequential learning strategies divide a large temporal domain into multiple subintervals and solve them one by one in a chronological order, which naturally respects the principle of causality and improves the stability of the PINN solution. The improved hard constraint strategy strictly ensures the continuity and smoothness of the PINN solution at time interval nodes, and at the same time passes the information from the previous interval to the next interval, which avoids the incorrect/trivial solution at the position far from the initial time. Furthermore, by investigating the requirements of different types of equations on hard constraints, we design a novel influence function with trainable parameters for hard constraints, which provides theoretical and technical support for the effective implementations of hard constraint strategies, and significantly improves the universality and computational accuracy of our method. In addition, an adaptive time-domain partitioning algorithm is proposed, which plays an important role in the application of the proposed method as well as in the improvement of computational efficiency and accuracy. Numerical experiments verify the performance of the method. The data and code accompanying this paper are available at https://github.com/zhizhi4452/HCS.
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References · 16
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- 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
- 2020Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning methodcited by 29
4 more
- 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
- 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
- 2023A practical PINN framework for multi-scale problems with multi-magnitude loss termscited by 15
- 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
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