# Deep latent force models: ODE-based process convolutions for Bayesian deep learning

## TL;DR

This paper introduces a new Bayesian deep learning model that uses physics-informed kernels to better capture dynamics in complex systems.

## Contribution

The novel deep latent force model (DLFM) integrates physics-informed kernels derived from ODEs into deep Gaussian processes.

## Key findings

- The DLFM effectively captures dynamics in nonlinear multi-output time series data.
- DLFM performs comparably to non-physics-informed models on regression tasks.
- Inducing points framework negatively impacts model extrapolation.

## Abstract

Modelling the behaviour of highly nonlinear dynamical systems with robust uncertainty quantification is a challenging task which typically requires approaches specifically designed to address the problem at hand. We introduce a domain-agnostic model to address this issue termed the deep latent force model (DLFM), a deep Gaussian process with physics-informed kernels at each layer, derived from ordinary differential equations using the framework of process convolutions. Two distinct formulations of the DLFM are presented which utilise weight-space and variational inducing points-based Gaussian process approximations, both of which are amenable to doubly stochastic variational inference. We present empirical evidence of the capability of the DLFM to capture the dynamics present in highly nonlinear real-world multi-output time series data. Additionally, we find that the DLFM is capable of achieving comparable performance to a range of non-physics-informed probabilistic models on benchmark univariate regression tasks. We also empirically assess the negative impact of the inducing points framework on the extrapolation capabilities of LFM-based models.