Physics-based polynomial neural networks for one-shot learning of dynamical systems from one or a few samples

TL;DR

This paper introduces a physics-informed polynomial neural network approach that leverages prior differential equation knowledge to enable effective one-shot learning of dynamical systems from minimal data, demonstrated on real experiments.

Contribution

It presents a novel method to initialize polynomial neural networks using Taylor mapping, enabling data-efficient learning of physical systems with limited samples.

Findings

Successfully applied to simple pendulum and X-ray source experiments

Recovered complex physics from noisy, limited, and partial data

Provided accurate predictions for unseen inputs

Abstract

This paper discusses an approach for incorporating prior physical knowledge into the neural network to improve data efficiency and the generalization of predictive models. If the dynamics of a system approximately follows a given differential equation, the Taylor mapping method can be used to initialize the weights of a polynomial neural network. This allows the fine-tuning of the model from one training sample of real system dynamics. The paper describes practical results on real experiments with both a simple pendulum and one of the largest worldwide X-ray source. It is demonstrated in practice that the proposed approach allows recovering complex physics from noisy, limited, and partial observations and provides meaningful predictions for previously unseen inputs. The approach mainly targets the learning of physical systems when state-of-the-art models are difficult to apply given the lack of training data.