A Three‐Stage PINN for Solving Inverse Problems of Piecewise‐Continuous Variable Coefficients of PDEs

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ABSTRACT In this paper, we propose a deep learning method called the three‐stage physics‐informed neural network (TS‐PINN) for solving inverse problems of piecewise‐continuous variable coefficients of partial differential equations (PDEs). Unlike the existing PINN‐type methods that typically use a composite loss functional to jointly optimize the solution and the coefficient, the novel TS‐PINN method employs two different neural networks: the solution and coefficient networks, and decomposes the complex training process into three stages according to all the information at hand. The solution network learns an initial approximation of the PDE solution in the first stage. Based on this approximation, the coefficient network estimates the unknown coefficients in the second stage. With the two networks learned in the first two stages, in the third stage, the two networks are trained together on newly constructed training sets. Our numerical tests show that the TS‐PINN is effective and accurate for various types of variable coefficients, including polynomial, trigonometric, exponential, space‐time dependent, and piecewise‐continuous functions. We also provide an in‐depth analysis of the TS‐PINN regarding the necessity of the three‐stage training and the anti‐noise analysis.

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