This study focuses on addressing the challenges of solving analytically intractable differential equations that arise in scientific and engineering fields such as Hamilton-Jacobi-Bellman. Traditional numerical methods and neural network approaches for solving such equations often require independent simulation or retraining when the underlying parameters change. To overcome this, this study employs a physics-informed DeepONet (PI-DeepONet) to approximate the solution operator of a nonlinear parabolic equation. PI-DeepONet integrates known physics into a deep neural network, which learns the solution of the PDE.

翻译：本研究聚焦于解决科学和工程领域（如Hamilton-Jacobi-Bellman方程）中出现的解析难以处理的微分方程所面临的挑战。传统的数值方法和神经网络方法在求解此类方程时，通常需要针对底层参数的变化进行独立仿真或重新训练。为了克服这一问题，本研究采用物理信息深度算子网络（PI-DeepONet）来近似非线性抛物型方程的解算子。PI-DeepONet将已知物理规律融入深度神经网络，从而学习偏微分方程的解。