Learning the parameters of a differential equation from its trajectory via the adjoint equation

TL;DR

This paper introduces a method using an extended adjoint equation to compute gradients for fitting differential equation parameters from measurements, bridging machine learning techniques with differential equations theory.

Contribution

It develops a novel framework for constructing loss functions and deriving gradients via an extended adjoint equation, enabling parameter fitting from noisy measurements.

Findings

Gradients derived from the extended adjoint equation enable effective parameter fitting.

The method works with both continuous and discrete noisy measurements.

Numerical experiments demonstrate successful application of gradient descent for parameter estimation.

Abstract

The paper contributes to strengthening the relation between machine learning and the theory of differential equations. In this context, the inverse problem of fitting the parameters, and the initial condition of a differential equation to some measurements constitutes a key issue. The paper explores an abstraction that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times.