Chris Budd

Andrew T. T. McRae, Colin J. Cotter, Chris J. Budd

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Ampère equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Ampère equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Ampère equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Ampère-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.

James Rowbottom, Georg Maierhofer, Teo Deveney et al.

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

Teo Deveney, Jan Stanczuk, Lisa Maria Kreusser et al.

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.

Simone Appella, Simon Arridge, Chris Budd et al.

We consider the problem of univariate nonlinear function approximation using shallow neural networks (NN) with a rectified linear unit (ReLU) activation function. We show that the $L_2$ based approximation problem is ill-conditioned and the behaviour of optimisation algorithms used in training these networks degrades rapidly as the width of the network increases. This can lead to significantly poorer approximation in practice than expected from the theoretical expressivity of the ReLU architecture and traditional methods such as univariate Free Knot Splines (FKS). Univariate shallow ReLU NNs and FKS span the same function space, and thus have the same theoretical expressivity. However, the FKS representation remains well-conditioned as the number of knots increases. We leverage the theory of optimal piecewise linear interpolants to improve the training procedure for ReLU NNs. Using the equidistribution principle, we propose a two-level procedure for training the FKS by first solving the nonlinear problem of finding the optimal knot locations of the interpolating FKS, and then determine the optimal weights and knots of the FKS by solving a nearly linear, well-conditioned problem. The training of the FKS gives insights into how we can train a ReLU NN effectively, with an equally accurate approximation. We combine the training of the ReLU NN with an equidistribution-based loss to find the breakpoints of the ReLU functions. This is then combined with preconditioning the ReLU NN approximation to find the scalings of the ReLU functions. This fast, well-conditioned and reliable method finds an accurate shallow ReLU NN approximation to a univariate target function. We test this method on a series of regular, singular, and rapidly varying target functions and obtain good results, realising the expressivity of the shallow ReLU network in all cases. We then extend our results to deeper networks.

Chaoyu Liu, Yangming Li, Zhongying Deng et al.

Fourier Neural Operators (FNOs) have recently emerged as a promising and efficient approach for learning the numerical solutions to partial differential equations (PDEs) from data. However, standard FNO often fails to preserve key conservation laws, such as mass conservation, momentum conservation, norm conservation, etc., which are crucial for accurately modeling physical systems. Existing methods for incorporating these conservation laws into Fourier neural operators are achieved by designing related loss function or incorporating post-processing method at the training time. None of them can both exactly and adaptively correct the outputs to satisfy conservation laws, and our experiments show that these methods can lead to inferior performance while preserving conservation laws. In this work, we propose a novel adaptive correction approach to ensure the conservation of fundamental quantities. Our method introduces a learnable matrix to adaptively adjust the solution to satisfy the conservation law during training. It ensures that the outputs exactly satisfy the goal conservation law and allow for more flexibility and adaptivity for the model to correct the outputs. We theoretically show that applying our adaptive correction to an unconstrained FNO yields a solution with data loss no worse than that of the best conservation-satisfying FNO. We compare our approach with existing methods on a range of representative PDEs. Experiment results show that our method consistently outperform other methods.

Chaoyu Liu, Chris Budd, Carola-Bibiane Schönlieb

Neural networks have emerged as promising tools for solving partial differential equations (PDEs), particularly through the application of neural operators. Training neural operators typically requires a large amount of training data to ensure accuracy and generalization. In this paper, we propose a novel data augmentation method specifically designed for training neural operators on evolution equations. Our approach utilizes insights from inverse processes of these equations to efficiently generate data from random initialization that are combined with original data. To further enhance the accuracy of the augmented data, we introduce high-order inverse evolution schemes. These schemes consist of only a few explicit computation steps, yet the resulting data pairs can be proven to satisfy the corresponding implicit numerical schemes. In contrast to traditional PDE solvers that require small time steps or implicit schemes to guarantee accuracy, our data augmentation method employs explicit schemes with relatively large time steps, thereby significantly reducing computational costs. Accuracy and efficacy experiments confirm the effectiveness of our approach. Additionally, we validate our approach through experiments with the Fourier Neural Operator and UNet on three common evolution equations that are Burgers' equation, the Allen-Cahn equation and the Navier-Stokes equation. The results demonstrate a significant improvement in the performance and robustness of the Fourier Neural Operator when coupled with our inverse evolution data augmentation method.