# Approximate Inference for Multiplicative Latent Force Models

## TL;DR

This paper introduces two approximate inference methods for multiplicative latent force models, enhancing their ability to model complex dynamic systems with controlled geometry, demonstrated on simulated and motion capture data.

## Contribution

It extends latent force models to include multiplicative interactions and proposes two novel inference techniques for this broader class.

## Key findings

- Both methods perform well on simulated data.
- The models effectively capture complex motion patterns.
- Application to motion capture data shows practical utility.

## Abstract

Latent force models are a class of hybrid models for dynamic systems, combining simple mechanistic models with flexible Gaussian process (GP) perturbations. An extension of this framework to include multiplicative interactions between the state and GP terms allows strong a priori control of the model geometry at the expense of tractable inference. In this paper we consider two methods of carrying out inference within this broader class of models. The first is based on an adaptive gradient matching approximation, and the second is constructed around mixtures of local approximations to the solution. We compare the performance of both methods on simulated data, and also demonstrate an application of the multiplicative latent force model on motion capture data.