Variational Autoencoding of PDE Inverse Problems

Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse problem, and can act as a drop-in regulariser that encodes a-priori physical information. We employ the variational form of the PDE problem and introduce stochastic local approximations as a form of model based data augmentation. We demonstrate both the accuracy and increased computational efficiency of the framework on real world settings and structured spatial processes.

Deep Orthogonal Decompositions for Convective Nowcasting

Near-term prediction of the structured spatio-temporal processes driving our climate is of profound importance to the safety and well-being of millions, but the prounced nonlinear convection of these processes make a complete mechanistic description even of the short-term dynamics challenging. However, convective transport provides not only a principled physical description of the problem, but is also indicative of the transport in time of informative features which has lead to the recent successful development of ``physics free'' approaches to the now-casting problem. In this work we demonstrate that their remains an important role to be played by physically informed models, which can successfully leverage deep learning (DL) to project the process onto a lower dimensional space on which a minimal dynamical description holds. Our approach synthesises the feature extraction capabilities of DL with physically motivated dynamics to outperform existing model free approaches, as well as state of the art hybrid approaches, on complex real world datasets including sea surface temperature and precipitation.

Approximate Inference for Multiplicative Latent Force Models

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.

Multiplicative Latent Force Models

Bayesian modelling of dynamic systems must achieve a compromise between providing a complete mechanistic specification of the process while retaining the flexibility to handle those situations in which data is sparse relative to model complexity, or a full specification is hard to motivate. Latent force models achieve this dual aim by specifying a parsimonious linear evolution equation which an additive latent Gaussian process (GP) forcing term. In this work we extend the latent force framework to allow for multiplicative interactions between the GP and the latent states leading to more control over the geometry of the trajectories. Unfortunately inference is no longer straightforward and so we introduce an approximation based on the method of successive approximations and examine its performance using a simulation study.