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 issue, 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 a NN to simultaneously approximate known observations and conform to the constraints of DEs. By utilizing the homotopy continuation method, the approximation traces the observations to identify multiple solutions and solve the inverse problem. 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：由于多解性、对称性和解空间中的分岔导致的复杂行为，求解具有多解的非线性微分方程反问题是一项具有挑战性的任务。为解决此问题，我们提出同伦物理信息神经网络（HomPINNs）——一种利用同伦延拓和神经网络（NNs）求解反问题的新框架。该框架首先使用神经网络同时逼近已知观测值并满足微分方程约束条件。通过同伦延拓方法，该近似沿观测值轨迹追踪以识别多解并求解反问题。实验部分在一维微分方程上测试所提方法的性能，并将其应用于二维Gray-Scott模拟。研究结果表明，该方法具有良好的可扩展性和适应性，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，例如复杂系统建模及物理、化学、生物学中的反问题求解。