Physics-informed machine learning (PIML) has established itself as a new scientific paradigm which enables the seamless integration of observational data with partial differential equation (PDE) based physics models. A powerful tool for the analysis, reduction and solution of PDEs is the Lie symmetry method. Nevertheless, only recently has the integration of such symmetries into PIML frameworks begun to be explored. The present work adds to this growing literature by introducing an approach for incorporating a Lie symmetry into a physics-informed Gaussian process (GP) model. The symmetry is introduced as a constraint on the GP; either in a soft manner via virtual observations of an induced PDE called the invariant surface condition, or explicitly through the design of the kernel. Experimental results demonstrate that the use of symmetry constraints improves the performance of the GP for both forward and inverse problems, and that our approach offers competitive performance with neural networks in the low-data environment.

We introduce a framework for designing boundary constrained Gaussian process (BCGP) priors for exact enforcement of linear boundary conditions, and apply it to the machine learning of (initial) boundary value problems involving linear partial differential equations (PDEs).In contrast to existing work, we illustrate how to design boundary constrained mean and kernel functions for all classes of boundary conditions typically used in PDE modelling, namely Dirichlet, Neumann, Robin and mixed conditions. Importantly, this is done in a manner which allows for both forward and inverse problems to be naturally accommodated. We prove that the BCGP kernel has a universal representational capacity under Dirichlet conditions, and establish a formal equivalence between BCGPs and boundary-constrained neural networks (BCNNs) of infinite width.Finally, extensive numerical experiments are performed involving several linear PDEs, the results of which demonstrate the effectiveness and robustness of BCGP inference in the presence of sparse, noisy data.

Combining biomechanical modelling of left ventricular (LV) function and dysfunction with cardiac magnetic resonance (CMR) imaging has the potential to improve the prognosis of patient-specific cardiovascular disease risks. Biomechanical studies of LV function in three dimensions usually rely on a computerized representation of the LV geometry based on finite element discretization, which is essential for numerically simulating in vivo cardiac dynamics. Detailed knowledge of the LV geometry is also relevant for various other clinical applications, such as assessing the LV cavity volume and wall thickness. Accurately and automatically reconstructing personalized LV geometries from conventional CMR images with minimal manual intervention is still a challenging task, which is a pre-requisite for any subsequent automated biomechanical analysis. We propose a deep learning-based automatic pipeline for predicting the three-dimensional LV geometry directly from routinely-available CMR cine images, without the need to manually annotate the ventricular wall. Our framework takes advantage of a low-dimensional representation of the high-dimensional LV geometry based on principal component analysis. We analyze how the inference of myocardial passive stiffness is affected by using our automatically generated LV geometries instead of manually generated ones. These insights will inform the development of statistical emulators of LV dynamics to avoid computationally expensive biomechanical simulations. Our proposed framework enables accurate LV geometry reconstruction, outperforming previous approaches by delivering a reconstruction error 50% lower than reported in the literature. We further demonstrate that for a nonlinear cardiac mechanics model, using our reconstructed LV geometries instead of manually extracted ones only moderately affects the inference of passive myocardial stiffness described by an anisotropic hyperelastic constitutive law. The developed methodological framework has the potential to make an important step towards personalized medicine by eliminating the need for time consuming and costly manual operations. In addition, our method automatically maps the CMR scan into a low-dimensional representation of the LV geometry, which constitutes an important stepping stone towards the development of an LV geometry-heterogeneous emulator.