High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networks

Time-dependent Partial Differential Equations (PDEs) arise frequently in reallife applications: e.g., diffusion process of liquid flows (see, e.g., [18]), heat distribution in time (see, e.g., [21]), simulations of nonlinear wave dynamics (see, e.g., [28]), groundwater flow dynamics (see, e.g., [4]), quantum dynamics (see, e.g., [8]), computational mechanics (see, e.g., [15]), etc. These applications are very important from both theoretical and practical points of view. For instance, groundwater flow simulations can be used to predict hydro-geological risks, which are crucial for infrastructures located in seismic or unstable regions (see, e.g., [1, 14]. High precision efficient simulations and modeling in this case can be used to predict different risks arising in this field. In this case, numerical models can be used to describe fluid dynamics: e.g., diffusion equation or Burgers' equations (see, e.g., [2]). In order to solve difficult nonlinear PDEs, there exist different approaches: e.g., finite element method (FEM, see, e.g., [13]) or Isogeometric analysis (IGA,