Partial differential equations (PDEs) are among the most ubiquitous tools in modeling natural phenomena, and various numerical methods have been developed for decades to solve PDE problems. While deep learning has introduced new techniques to the field, the limited accuracy of deep neural networks hinders their application to scientific problems that require high precision.

This talk will present a warm-start approach that combines the strengths of deep neural networks and classical numerical solvers. The approach uses neural networks to provide an initial guess, enabling classical numerical solvers to achieve a good solution more efficiently. We will demonstrate the advantages of the proposed method through two examples. In the first example, we will demonstrate how the initial guess provided by neural networks helps identify the basin of attraction of the true solution in the inverse scattering problem. In the second example, we will show how the initial guess provided by neural networks leads to faster convergence in solving the Reynolds-averaged Navier-Stokes equations. In both examples, the combination of the classical PDE solver and the neural network outperforms either approach alone. The potential of this approach will be discussed, along with the new challenges that must be tackled to solve challenging scientific problems using this new paradigm.