Abstract:
Simulation and inference are cornerstones of scientific inquiry.
Numerous domains and applications can integrate advanced machine-learning methods for reasoning from prior mechanistic knowledge and data.
Probabilistic approaches take into account the uncertainty associated with observations and the physical model.
Bayesian inference, however, remains challenging in practice.
In particular, high-dimensional settings that revolve around spatiotemporal dynamical systems, such as atmospheric models for predicting weather and climate, are computationally demanding.
This thesis considers two different avenues for integrating large physical systems into a Bayesian inference pipeline.
On the one hand, assuming linear dynamics governing the latent system state and capturing mechanistic and empirical uncertainty in tractable Gaussian noise models allows optimal state estimation via the Kalman filter.
On the other hand, generative diffusion models that learn probabilistic representations of complex nonlinear dynamics from existing trajectories can be leveraged as prior models in Bayesian inverse problems.
For both approaches, open conceptual and practical questions are addressed.
High-dimensional state spaces create a gap between the theoretical merits of the Kalman filter and its practicability.
Concretely, the explicit modeling of pairwise errors that characterizes Gaussian models impedes their application to large-scale systems due to limited memory and compute.
Low-dimensional representations of the pairwise error covariances aim to resolve this dilemma.
Sampling-based approaches are broadly applicable but introduce stochasticity into the inference algorithm.
For linear systems with a Gauss--Markov process-noise model, we develop a fully deterministic alternative.
The proposed algorithm formulates prediction, update, and smoothing routines for optimal low-rank approximations of the covariance matrices and converges to the exact Kalman filter in the full-rank limit.
Especially in scientific applications, imposing mechanistic constraints on deep-learning models increasingly blurs the line between inference and simulation.
Recent research on generative diffusion models establishes the model class as powerful statistical representations of physical processes.
Their probabilistic nature and post-training conditioning make diffusion models suitable priors in flexible Bayesian inference models.
We investigate and demonstrate this approach in the context of downscaling climate simulations.
The proposed generative downscaling framework generates spatiotemporally coherent trajectories that estimate fine-scale processes obscured by simulating climate on coarse grids.
Experiments on coarse weather and climate input illustrate that the sampled trajectories accurately estimate several atmospheric variables, simultaneously, on a fine spatiotemporal grid.
Sampling multiple predictions provides structured uncertainty.
Lastly, we use this generative downscaling framework for local risk assessment of renewable-energy shortages based on existing simulations of future climate.
machine learning