ArticleJournal of computational physics2024
HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions.
Article in Journal of computational physics, 2024. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 7 papers.
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Who cites it
7 citing papers in PubMed.
- PatternFormer: Learning Multiple Solution Patterns in Reaction-Diffusion Systems.Research square · 2026Article
- Inferring neural sources from electroencephalography: foundations and frontiers.Journal of neural engineering · 2026Review
- ZENN: A thermodynamics-inspired computational framework for heterogeneous data-driven modeling.Proceedings of the National Academy of Sciences of the United States of America · 2026Article
- On pattern formation in the thermodynamically-consistent variational Gray-Scott model.Mathematical biosciences · 2025Article
- Research on temperature prediction model of molten steel of tundish in continuous casting.Scientific reports · 2024Article
- Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics.Applied mathematics and mechanics · 2023Article
- Constructing Physics-Informed Neural Networks with Architecture Based on Analytical Modification of Numerical Methods by Solving the Problem of Modelling Processes in a Chemical Reactor.Sensors (Basel, Switzerland) · 2023Article
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5 authors.
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Abstract
Due to the complex behavior arising from non-uniqueness, symmetry, and bifurcations in the solution space, solving inverse problems of nonlinear differential equations (DEs) with multiple solutions is a challenging task. To address this, we propose homotopy physics-informed neural networks (HomPINNs), a novel framework that leverages homotopy continuation and neural networks (NNs) to solve inverse problems. The proposed framework begins with the use of NNs to simultaneously approximate unlabeled observations across diverse solutions while adhering to DE constraints. Through homotopy continuation, the proposed method solves the inverse problem by tracing the observations and identifying multiple solutions. The experiments involve testing the performance of the proposed method on one-dimensional DEs and applying it to solve a two-dimensional Gray-Scott simulation. Our findings demonstrate that the proposed method is scalable and adaptable, providing an effective solution for solving DEs with multiple solutions and unknown parameters. Moreover, it has significant potential for various applications in scientific computing, such as modeling complex systems and solving inverse problems in physics, chemistry, biology, etc.
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