Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems

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.

Plasma Surrogate Modelling using Fourier Neural Operators

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.

Gaussian Processes on Cellular Complexes

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.

Thin and Deep Gaussian Processes

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.

A Unifying Variational Framework for Gaussian Process Motion Planning

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.

Faster Training of Neural ODEs Using Gauß-Legendre Quadrature

Neural ODEs demonstrate strong performance in generative and time-series modelling. However, training them via the adjoint method is slow compared to discrete models due to the requirement of numerically solving ODEs. To speed neural ODEs up, a common approach is to regularise the solutions. However, this approach may affect the expressivity of the model; when the trajectory itself matters, this is particularly important. In this paper, we propose an alternative way to speed up the training of neural ODEs. The key idea is to speed up the adjoint method by using Gau{\ss}-Legendre quadrature to solve integrals faster than ODE-based methods while remaining memory efficient. We also extend the idea to training SDEs using the Wong-Zakai theorem, by training a corresponding ODE and transferring the parameters. Our approach leads to faster training of neural ODEs, especially for large models. It also presents a new way to train SDE-based models.

Grasp Transfer based on Self-Aligning Implicit Representations of Local Surfaces

Objects we interact with and manipulate often share similar parts, such as handles, that allow us to transfer our actions flexibly due to their shared functionality. This work addresses the problem of transferring a grasp experience or a demonstration to a novel object that shares shape similarities with objects the robot has previously encountered. Existing approaches for solving this problem are typically restricted to a specific object category or a parametric shape. Our approach, however, can transfer grasps associated with implicit models of local surfaces shared across object categories. Specifically, we employ a single expert grasp demonstration to learn an implicit local surface representation model from a small dataset of object meshes. At inference time, this model is used to transfer grasps to novel objects by identifying the most geometrically similar surfaces to the one on which the expert grasp is demonstrated. Our model is trained entirely in simulation and is evaluated on simulated and real-world objects that are not seen during training. Evaluations indicate that grasp transfer to unseen object categories using this approach can be successfully performed both in simulation and real-world experiments. The simulation results also show that the proposed approach leads to better spatial precision and grasp accuracy compared to a baseline approach.

Neural Field Movement Primitives for Joint Modelling of Scenes and Motions

This paper presents a novel Learning from Demonstration (LfD) method that uses neural fields to learn new skills efficiently and accurately. It achieves this by utilizing a shared embedding to learn both scene and motion representations in a generative way. Our method smoothly maps each expert demonstration to a scene-motion embedding and learns to model them without requiring hand-crafted task parameters or large datasets. It achieves data efficiency by enforcing scene and motion generation to be smooth with respect to changes in the embedding space. At inference time, our method can retrieve scene-motion embeddings using test time optimization, and generate precise motion trajectories for novel scenes. The proposed method is versatile and can employ images, 3D shapes, and any other scene representations that can be modeled using neural fields. Additionally, it can generate both end-effector positions and joint angle-based trajectories. Our method is evaluated on tasks that require accurate motion trajectory generation, where the underlying task parametrization is based on object positions and geometric scene changes. Experimental results demonstrate that the proposed method outperforms the baseline approaches and generalizes to novel scenes. Furthermore, in real-world experiments, we show that our method can successfully model multi-valued trajectories, it is robust to the distractor objects introduced at inference time, and it can generate 6D motions.

Sliding Touch-based Exploration for Modeling Unknown Object Shape with Multi-fingered Hands

Efficient and accurate 3D object shape reconstruction contributes significantly to the success of a robot's physical interaction with its environment. Acquiring accurate shape information about unknown objects is challenging, especially in unstructured environments, e.g. the vision sensors may only be able to provide a partial view. To address this issue, tactile sensors could be employed to extract local surface information for more robust unknown object shape estimation. In this paper, we propose a novel approach for efficient unknown 3D object shape exploration and reconstruction using a multi-fingered hand equipped with tactile sensors and a depth camera only providing a partial view. We present a multi-finger sliding touch strategy for efficient shape exploration using a Bayesian Optimization approach and a single-leader-multi-follower strategy for multi-finger smooth local surface perception. We evaluate our proposed method by estimating the 3D shape of objects from the YCB and OCRTOC datasets based on simulation and real robot experiments. The proposed approach yields successful reconstruction results relying on only a few continuous sliding touches. Experimental results demonstrate that our method is able to model unknown objects in an efficient and accurate way.