Neural Network Differential Equation Solvers allow unsupervised error estimation and correction

Akshunna S. Dogra, Jeffrey B. Lai, Min Chul Lee, Martin Peev

Keywords: Deep learning, Numerical Methods, Differential Equations

TL;DR: A blueprint for deep learning solution models for differential equation

Abstract: Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy interpretability, etc. However, most NN DE solvers suffer from a fundamental limitation: their loss functions are not explicitly dependent on the errors associated with the solution estimates. As such, validation and error estimation usually requires knowledge of the true solution. Indeed, when the true solution is unknown, we are often reduced to simply hoping that a ``\textit{low enough}'' loss implies ``\textit{small enough}'' errors, since explicit relationships between the two are not available. In this work, we describe a general strategy for efficiently constructing error estimates and corrections for Neural Network Differential Equation solvers. Our methods do not require \textit{a priori} knowledge of the true solutions and obtain explicit relationships between loss functions and the errors, given certain assumptions on the DE. In turn, these explicit relationships directly allow us to estimate and correct for the errors.

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