James Rowbottom

Sourya Basu, Jose Gallego-Posada, Francesco Viganò et al. · mila

Equivariance to symmetries has proven to be a powerful inductive bias in deep learning research. Recent works on mesh processing have concentrated on various kinds of natural symmetries, including translations, rotations, scaling, node permutations, and gauge transformations. To date, no existing architecture is equivariant to all of these transformations. In this paper, we present an attention-based architecture for mesh data that is provably equivariant to all transformations mentioned above. Our pipeline relies on the use of relative tangential features: a simple, effective, equivariance-friendly alternative to raw node positions as inputs. Experiments on the FAUST and TOSCA datasets confirm that our proposed architecture achieves improved performance on these benchmarks and is indeed equivariant, and therefore robust, to a wide variety of local/global transformations.

Francesco Di Giovanni, James Rowbottom, Benjamin P. Chamberlain et al.

Graph Neural Networks (GNNs) typically operate by message-passing, where the state of a node is updated based on the information received from its neighbours. Most message-passing models act as graph convolutions, where features are mixed by a shared, linear transformation before being propagated over the edges. On node-classification tasks, graph convolutions have been shown to suffer from two limitations: poor performance on heterophilic graphs, and over-smoothing. It is common belief that both phenomena occur because such models behave as low-pass filters, meaning that the Dirichlet energy of the features decreases along the layers incurring a smoothing effect that ultimately makes features no longer distinguishable. In this work, we rigorously prove that simple graph-convolutional models can actually enhance high frequencies and even lead to an asymptotic behaviour we refer to as over-sharpening, opposite to over-smoothing. We do so by showing that linear graph convolutions with symmetric weights minimize a multi-particle energy that generalizes the Dirichlet energy; in this setting, the weight matrices induce edge-wise attraction (repulsion) through their positive (negative) eigenvalues, thereby controlling whether the features are being smoothed or sharpened. We also extend the analysis to non-linear GNNs, and demonstrate that some existing time-continuous GNNs are instead always dominated by the low frequencies. Finally, we validate our theoretical findings through ablations and real-world experiments.

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.

Niccolò Grillo, James Rowbottom, Pietro Liò et al.

Adaptive mesh refinement is central to the efficient solution of partial differential equations (PDEs) via the finite element method (FEM). Classical $r$-adaptivity optimizes vertex positions but requires solving expensive auxiliary PDEs such as the Monge-Ampère equation, while classical $h$-adaptivity modifies topology through element subdivision but suffers from expensive error indicator computation and is constrained by isotropic refinement patterns that impose accuracy ceilings. Combined $hr$-adaptive techniques naturally outperform single-modality approaches, yet inherit both computational bottlenecks and the restricted cost-accuracy trade-off. Emerging machine learning methods for adaptive mesh refinement seek to overcome these limitations, but existing approaches address $h$-adaptivity or $r$-adaptivity in isolation. We present HypeR, a deep reinforcement learning framework that jointly optimizes mesh relocation and refinement. HypeR casts the joint adaptation problem using tools from hypergraph neural networks and multi-agent reinforcement learning. Refinement is formulated as a heterogeneous multi-agent Markov decision process (MDP) where element agents decide discrete refinement actions, while relocation follows an anisotropic diffusion-based policy on vertex agents with provable prevention of mesh tangling. The reward function combines local and global error reduction to promote general accuracy. Across benchmark PDEs, HypeR reduces approximation error by up to 6--10$\times$ versus state-of-art $h$-adaptive baselines at comparable element counts, breaking through the uniform refinement accuracy ceiling that constrains subdivision-only methods. The framework produces meshes with improved shape metrics and alignment to solution anisotropy, demonstrating that jointly learned $hr$-adaptivity strategies can substantially enhance the capabilities of automated mesh generation.

William Lauga, James Rowbottom, Alexander Denker et al.

We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme that alternates between FEM-based inversion and learned graph neural regularization. The forward problem is numerically solved using the finite element method (FEM), enabling applicability to a wide range of geometries and PDEs. By leveraging the graph structure inherent to FEM discretizations, we employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches. Numerical experiments demonstrate that our framework outperforms classical regularization techniques and achieves accurate reconstructions even in highly ill-posed scenarios.

T. Konstantin Rusch, Benjamin P. Chamberlain, James Rowbottom et al.

We propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear controlled and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e.g. convolutional or attentional) as the coupling function, from which a multi-layer deep neural network is built up via the dynamics of the proposed ODEs. We relate the oversmoothing problem, commonly encountered in GNNs, to the stability of steady states of the underlying ODE and show that zero-Dirichlet energy steady states are not stable for our proposed ODEs. This demonstrates that the proposed framework mitigates the oversmoothing problem. Moreover, we prove that GraphCON mitigates the exploding and vanishing gradients problem to facilitate training of deep multi-layer GNNs. Finally, we show that our approach offers competitive performance with respect to the state-of-the-art on a variety of graph-based learning tasks.