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.

翻译：由于解空间中非唯一性、对称性和分岔引起的复杂行为，求解具有多解的非线性微分方程逆问题是一项具有挑战性的任务。为此，我们提出了同伦物理信息神经网络（HomPINNs），这是一种利用同伦延拓和神经网络求解逆问题的新型框架。该框架首先使用神经网络在满足微分方程约束的同时，同时逼近跨不同解的无标注观测数据。通过同伦延拓，所提方法通过追踪观测数据并识别多个解来求解逆问题。实验部分测试了该方法在一维微分方程上的性能，并将其应用于求解二维Gray-Scott模拟。研究结果表明，所提方法具有可扩展性和适应性，为求解具有多解和未知参数的微分方程提供了有效方案。此外，该方法在科学计算领域具有广泛的应用潜力，例如物理、化学、生物学等领域的复杂系统建模与逆问题求解。