# Deep Relaxation: partial differential equations for optimizing deep neural networks

## TL;DR

This paper links non-convex neural network training methods to nonlinear PDEs, showing that relaxation techniques can improve optimization performance and provide insights into the energy landscape of deep learning.

## Contribution

It introduces a PDE-based relaxation approach for neural network optimization, offering a novel theoretical framework and demonstrating improved performance over traditional methods.

## Key findings

- Relaxation techniques interpreted as viscous Hamilton-Jacobi PDE solutions.
- The PDE approach outperforms stochastic gradient descent in expectation.
- Analysis of the energy landscape confirms empirical observations.

## Abstract

In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already been used successfully in this context are reinterpreted as solutions of a viscous Hamilton-Jacobi PDE. Using a stochastic control interpretation allows we prove that the modified algorithm performs better in expectation that stochastic gradient descent. Well-known PDE regularity results allow us to analyze the geometry of the relaxed energy landscape, confirming empirical evidence. The PDE is derived from a stochastic homogenization problem, which arises in the implementation of the algorithm. The algorithms scale well in practice and can effectively tackle the high dimensionality of modern neural networks.