Neural PDEs offer efficient alternatives to computationally expensive numerical PDE solvers for simulating complex physical systems. However, their lack of robust uncertainty quantification (UQ) limits deployment in critical applications. We introduce a model-agnostic, physics-informed conformal prediction (CP) framework that provides guaranteed uncertainty estimates without requiring labelled data. By utilising a physics-based approach, we are able to quantify and calibrate the model's inconsistencies with the PDE rather than the uncertainty arising from the data. Our approach uses convolutional layers as finite-difference stencils and leverages physics residual errors as nonconformity scores, enabling data-free UQ with marginal and joint coverage guarantees across prediction domains for a range of complex PDEs. We further validate the efficacy of our method on neural PDE models for plasma modelling and shot design in fusion reactors.

Foundation models (FMs) are pre-trained on large-scale datasets and then fine-tuned on a downstream task for a specific application. The most successful and most commonly used fine-tuning method is to update the pre-trained weights via a low-rank adaptation (LoRA). LoRA introduces new weight matrices that are usually initialized at random with a uniform rank distribution across the model weights. Recent works focus on different initialization schemes or the learning of adaptive ranks during fine-tuning. Both approaches have only been investigated in isolation, resulting in slow convergence or a uniform rank distribution, in turn leading to suboptimal performance. We propose to improve LoRA by initializing the new weights in a data-driven manner by computing singular value decomposition (SVD) on minibatches of activation vectors. Then, we initialize the LoRA matrices with the obtained right-singular vectors and redistribute ranks among all weight matrices to provably store the maximum amount of information of the downstream data in the newly introduced weights. In this way, only what information to maintain or neglect during the fine-tuning process needs to be learned. We call our new method $\textbf{E}$xplained $\textbf{V}$ariance $\textbf{A}$daptation (EVA). We apply EVA to a variety of fine-tuning tasks ranging from language generation and understanding to image classification and reinforcement learning. EVA exhibits faster convergence than competitors and achieves the highest average score across a multitude of tasks per domain while reducing the number of trainable parameters through rank redistribution.

Soft robots offer more flexibility, compliance, and adaptability than traditional rigid robots. They are also typically lighter and cheaper to manufacture. However, their use in real-world applications is limited due to modeling challenges and difficulties in integrating effective proprioceptive sensors. Large-scale soft robots ($\approx$ two meters in length) have greater modeling complexity due to increased inertia and related effects of gravity. Common efforts to ease these modeling difficulties such as assuming simple kinematic and dynamics models also limit the general capabilities of soft robots and are not applicable in tasks requiring fast, dynamic motion like throwing and hammering. To overcome these challenges, we propose a data-efficient Bayesian optimization-based approach for learning control policies for dynamic tasks on a large-scale soft robot. Our approach optimizes the task objective function directly from commanded pressures, without requiring approximate kinematics or dynamics as an intermediate step. We demonstrate the effectiveness of our approach through both simulated and real-world experiments.

In recent years, machine learning has established itself as a powerful tool for high-resolution weather forecasting. While most current machine learning models focus on deterministic forecasts, accurately capturing the uncertainty in the chaotic weather system calls for probabilistic modeling. We propose a probabilistic weather forecasting model called Graph-EFM, combining a flexible latent-variable formulation with the successful graph-based forecasting framework. The use of a hierarchical graph construction allows for efficient sampling of spatially coherent forecasts. Requiring only a single forward pass per time step, Graph-EFM allows for fast generation of arbitrarily large ensembles. We experiment with the model on both global and limited area forecasting. Ensemble forecasts from Graph-EFM achieve equivalent or lower errors than comparable deterministic models, with the added benefit of accurately capturing forecast uncertainty.

Data assimilation is a core component of numerical weather prediction systems. The large quantity of data processed during assimilation requires the computation to be distributed across increasingly many compute nodes, yet existing approaches suffer from synchronisation overhead in this setting. In this paper, we exploit the formulation of data assimilation as a Bayesian inference problem and apply a message-passing algorithm to solve the spatial inference problem. Since message passing is inherently based on local computations, this approach lends itself to parallel and distributed computation. In combination with a GPU-accelerated implementation, we can scale the algorithm to very large grid sizes while retaining good accuracy and compute and memory requirements.

Data assimilation (DA) methods use priors arising from differential equations to robustly interpolate and extrapolate data. Popular techniques such as ensemble methods that handle high-dimensional, nonlinear PDE priors focus mostly on state estimation, however can have difficulty learning the parameters accurately. On the other hand, machine learning based approaches can naturally learn the state and parameters, but their applicability can be limited, or produce uncertainties that are hard to interpret. Inspired by the Integrated Nested Laplace Approximation (INLA) method in spatial statistics, we propose an alternative approach to DA based on iteratively linearising the dynamical model. This produces a Gaussian Markov random field at each iteration, enabling one to use INLA to infer the state and parameters. Our approach can be used for arbitrary nonlinear systems, while retaining interpretability, and is furthermore demonstrated to outperform existing methods on the DA task. By providing a more nuanced approach to handling nonlinear PDE priors, our methodology offers improved accuracy and robustness in predictions, especially where data sparsity is prevalent.

Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion. Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors. Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models. We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier Neural Operators (FNO). We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (MSE $\approx$ $10^{-5}$). Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations (PDE), and can capture the dependence among the different variables in a single model. FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e., cameras looking across the central solenoid and the divertor in the Tokamak. We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring. We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full duration of the plasma shot within MAST. The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.

In recent years, there has been considerable interest in developing machine learning models on graphs in order to account for topological inductive biases. In particular, recent attention was given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.

Gaussian processes (GPs) can provide a principled approach to uncertainty quantification with easy-to-interpret kernel hyperparameters, such as the lengthscale, which controls the correlation distance of function values. However, selecting an appropriate kernel can be challenging. Deep GPs avoid manual kernel engineering by successively parameterizing kernels with GP layers, allowing them to learn low-dimensional embeddings of the inputs that explain the output data. Following the architecture of deep neural networks, the most common deep GPs warp the input space layer-by-layer but lose all the interpretability of shallow GPs. An alternative construction is to successively parameterize the lengthscale of a kernel, improving the interpretability but ultimately giving away the notion of learning lower-dimensional embeddings. Unfortunately, both methods are susceptible to particular pathologies which may hinder fitting and limit their interpretability. This work proposes a novel synthesis of both previous approaches: Thin and Deep GP (TDGP). Each TDGP layer defines locally linear transformations of the original input data maintaining the concept of latent embeddings while also retaining the interpretation of lengthscales of a kernel. Moreover, unlike the prior solutions, TDGP induces non-pathological manifolds that admit learning lower-dimensional representations. We show with theoretical and experimental results that i) TDGP is, unlike previous models, tailored to specifically discover lower-dimensional manifolds in the input data, ii) TDGP behaves well when increasing the number of layers, and iii) TDGP performs well in standard benchmark datasets.

To control how a robot moves, motion planning algorithms must compute paths in high-dimensional state spaces while accounting for physical constraints related to motors and joints, generating smooth and stable motions, avoiding obstacles, and preventing collisions. A motion planning algorithm must therefore balance competing demands, and should ideally incorporate uncertainty to handle noise, model errors, and facilitate deployment in complex environments. To address these issues, we introduce a framework for robot motion planning based on variational Gaussian Processes, which unifies and generalizes various probabilistic-inference-based motion planning algorithms. Our framework provides a principled and flexible way to incorporate equality-based, inequality-based, and soft motion-planning constraints during end-to-end training, is straightforward to implement, and provides both interval-based and Monte-Carlo-based uncertainty estimates. We conduct experiments using different environments and robots, comparing against baseline approaches based on the feasibility of the planned paths, and obstacle avoidance quality. Results show that our proposed approach yields a good balance between success rates and path quality.