From {\tt Ferminet} to PINN. Connections between neural network-based algorithms for high-dimensional Schr\"odinger Hamiltonian

TL;DR

This paper explores the connections between neural network algorithms used for solving high-dimensional Schrödinger equations across applied mathematics, engineering, and quantum chemistry, highlighting reformulations and shared optimization strategies.

Contribution

It establishes links between neural network PDE solvers like PINN and quantum chemistry algorithms such as Ferminet, offering new perspectives on their underlying optimization methods.

Findings

Reformulation of PINN as a data fitting problem.

Connection between neural network-based VMC and Diffusion Monte Carlo.

Shared optimization strategies across different algorithm frameworks.

Abstract

In this note, we establish some connections between standard (data-driven) neural network-based solvers for PDE and eigenvalue problems developed on one side in the applied mathematics and engineering communities (e.g. Deep-Ritz and Physics Informed Neural Networks (PINN)), and on the other side in quantum chemistry (e.g. Variational Monte Carlo algorithms, {\tt Ferminet} or {\tt Paulinet} following the pioneer work of {\it Carleo et. al}. In particular, we re-formulate a PINN algorithm as a {\it fitting} problem with data corresponding to the solution to a standard Diffusion Monte Carlo algorithm initialized thanks to neural network-based Variational Monte Carlo. Connections at the level of the optimization algorithms are also established.