Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.

Deep surrogate accelerated delayed-acceptance HMC for Bayesian inference of spatio-temporal heat fluxes in rotating disc systems

We study the Bayesian inverse problem of inferring the Biot number, a spatio-temporal heat-flux parameter in a PDE model. This is an ill-posed problem where standard optimisation yields unphysical inferences. We introduce a training scheme that uses temperature data to adaptively train a neural-network surrogate to simulate the parametric forward model. This approach approximates forward and inverse solution together, by simultaneously identifying an approximate posterior distribution over the Biot number, and weighting the forward training loss according to this approximation. Utilising random Chebyshev series, we outline how to approximate an arbitrary Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to efficiently sample from the corresponding posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss function approaches zero. Furthermore, we describe how this surrogate-accelerated HMC approach can be combined with a traditional PDE solver in a delayed-acceptance scheme to a-priori control the posterior accuracy, thus overcoming a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high-dimensional parameter spaces, whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results compare the accuracy and efficiency of the adaptive and general training regimes, as well as various Markov chain Monte Carlo proposals strategies.

A deep surrogate approach to efficient Bayesian inversion in PDE and integral equation models

We propose a novel deep learning approach to efficiently perform Bayesian inference in partial differential equation (PDE) and integral equation models over potentially high-dimensional parameter spaces. The contributions of this paper are two-fold; the first is the introduction of a neural network approach to approximating the solutions of Fredholm and Volterra integral equations of the first and second kind. The second is the description of a deep surrogate model which allows for efficient sampling from a Bayesian posterior distribution in which the likelihood depends on the solutions of PDEs or integral equations. For the latter, our method relies on the approximation of parametric solutions by neural networks. This deep learning approach allows for parametric solutions to be approximated accurately in significantly higher dimensions than is possible using classical techniques. These solutions are very cheap to evaluate, making Bayesian inference over large parameter spaces tractable for these models using Markov chain Monte Carlo. We demonstrate this method using two real-world examples; these include Bayesian inference in the PDE and integral equation case for an example from electrochemistry, and Bayesian inference of a function-valued heat-transfer parameter with applications in aviation.