Data-Free Deep Neural Networks for Solving Partial Differential Equations in Nanobiophysics

Partial differential equations (PDEs) in nanobiophysics (NBP) often arise in complicated geometries. Typically, such problems would be solved with mesh-based numerical solvers. However, the stochastic many-body systems common to NBP are described by high-dimensional PDEs. Mesh-based solvers fail for such problems, so instead these PDEs are solved indirectly using particle simulations. Still, these particles are often subject to force fields, which are themselves described by similar PDEs. Furthermore, to establish how observables, like molecular mobility, depend on problem parameters, like molecular size, simulations must be repeated many times.

A new method for solving PDEs is to approximate solutions with deep neural networks (DNNs). DNNs can even learn solutions directly from the PDE problem statement, without using any external data. In this talk, I will illustrate some benefits of this method for solving PDEs in NBP. DNNs are memory-efficient, enabling complicated electric fields to be used in GPU-accelerated particle simulations. Surprisingly, DNNs can actually solve high-dimensional PDEs directly, as an alternative to particle simulations. Finally, this method can naturally be extended to express target observables as differentiable functions of problem parameters.