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Due to the complex behavior arising from non-uniqueness, symmetry, andbifurcations in the solution space, solving inverse problems of nonlineardifferential equations (DEs) with multiple solutions is a challenging task. Toaddress 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 ofNNs to simultaneously approximate unlabeled observations across diversesolutions while adhering to DE constraints. Through homotopy continuation, theproposed method solves the inverse problem by tracing the observations andidentifying multiple solutions. The experiments involve testing the performanceof the proposed method on one-dimensional DEs and applying it to solve atwo-dimensional Gray-Scott simulation. Our findings demonstrate that theproposed method is scalable and adaptable, providing an effective solution forsolving DEs with multiple solutions and unknown parameters. Moreover, it hassignificant potential for various applications in scientific computing, such asmodeling complex systems and solving inverse problems in physics, chemistry,biology, etc.

HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions