Gefan Yang

Gefan Yang, Stefan Sommer

We propose a novel denoising diffusion generative model for predicting nonlinear fluid fields named FluidDiff. By performing a diffusion process, the model is able to learn a complex representation of the high-dimensional dynamic system, and then Langevin sampling is used to generate predictions for the flow state under specified initial conditions. The model is trained with finite, discrete fluid simulation data. We demonstrate that our model has the capacity to model the distribution of simulated training data and that it gives accurate predictions on the test data. Without encoded prior knowledge of the underlying physical system, it shares competitive performance with other deep learning models for fluid prediction, which is promising for investigation on new computational fluid dynamics methods.

Stefan Sommer, Gefan Yang, Elizabeth Louise Baker

Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.

Gefan Yang, Frank van der Meulen, Stefan Sommer

Inference in non-linear continuous stochastic processes on trees is challenging, particularly when observations are sparse (leaf-only) and the topology is complex. Exact smoothing via Doob's $h$-transform is intractable for general non-linear dynamics, while particle-based methods degrade in high dimensions. We propose Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for both discrete transitions and continuous diffusions. Our method constructs a variational posterior by leveraging an auxiliary linear-Gaussian process. This auxiliary process yields a closed-form backward filter that serves as a ``guide'', steering the generative path toward high-likelihood regions. We then learn a neural residual--parameterized as a normalizing flow or a controlled SDE--to capture the non-linear discrepancies. This formulation allows for an unbiased path-wise subsampling scheme, reducing the training complexity from tree-size dependent to path-length dependent. Empirical results show that NBFFG outperforms baselines on synthetic benchmarks, and we demonstrate the method on a high-dimensional inference task in phylogenetic analysis with reconstruction of ancestral butterfly wing shapes.

Elizabeth Louise Baker, Gefan Yang, Michael L. Severinsen et al.

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.

Nicklas Boserup, Gefan Yang, Michael Lind Severinsen et al.

We introduce a methodology for performing parameter inference in high-dimensional, non-linear diffusion processes. We illustrate its applicability for obtaining insights into the evolution of and relationships between species, including ancestral state reconstruction. Estimation is performed by utilising score matching to approximate diffusion bridges, which are subsequently used in an importance sampler to estimate log-likelihoods. The entire setup is differentiable, allowing gradient ascent on approximated log-likelihoods. This allows both parameter inference and diffusion mean estimation. This novel, numerically stable, score matching-based parameter inference framework is presented and demonstrated on biological two- and three-dimensional morphometry data.

Gefan Yang, Frank van der Meulen, Stefan Sommer

We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or score modeling. Compared to existing methods, it offers greater robustness across various diffusion specifications and conditioning scenarios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We introduce a flexible variational family, partially specified by a neural network, for approximating the diffusion bridge path measure. Once trained, it enables efficient sampling of independent bridges at a cost comparable to sampling the unconditioned (forward) process.