Towards Universal Mesh Movement Networks

Solving complex Partial Differential Equations (PDEs) accurately and efficiently is an essential and challenging problem in all scientific and engineering disciplines. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without increasing the overall mesh degree of freedom count. Conventional sophisticated mesh movement methods are extremely expensive and struggle to handle scenarios with complex boundary geometries. However, existing learning-based methods require re-training from scratch given a different PDE type or boundary geometry, which limits their applicability, and also often suffer from robustness issues in the form of inverted elements. In this paper, we introduce the Universal Mesh Movement Network (UM2N), which -- once trained -- can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries. UM2N consists of a Graph Transformer (GT) encoder for extracting features and a Graph Attention Network (GAT) based decoder for moving the mesh. We evaluate our method on advection and Navier-Stokes based examples, as well as a real-world tsunami simulation case. Our method outperforms existing learning-based mesh movement methods in terms of the benchmarks described above. In comparison to the conventional sophisticated Monge-Amp\`ere PDE-solver based method, our approach not only significantly accelerates mesh movement, but also proves effective in scenarios where the conventional method fails. Our project page is at https://erizmr.github.io/UM2N/.

End-to-end Wind Turbine Wake Modelling with Deep Graph Representation Learning

Wind turbine wake modelling is of crucial importance to accurate resource assessment, to layout optimisation, and to the operational control of wind farms. This work proposes a surrogate model for the representation of wind turbine wakes based on a state-of-the-art graph representation learning method termed a graph neural network. The proposed end-to-end deep learning model operates directly on unstructured meshes and has been validated against high-fidelity data, demonstrating its ability to rapidly make accurate 3D flow field predictions for various inlet conditions and turbine yaw angles. The specific graph neural network model employed here is shown to generalise well to unseen data and is less sensitive to over-smoothing compared to common graph neural networks. A case study based upon a real world wind farm further demonstrates the capability of the proposed approach to predict farm scale power generation. Moreover, the proposed graph neural network framework is flexible and highly generic and as formulated here can be applied to any steady state computational fluid dynamics simulations on unstructured meshes.

E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation

Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. However, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an 'enriched' approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a "data-driven" goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-by-element construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-by-element approach implies reasonably low training costs.

Learning to Estimate and Refine Fluid Motion with Physical Dynamics

Extracting information on fluid motion directly from images is challenging. Fluid flow represents a complex dynamic system governed by the Navier-Stokes equations. General optical flow methods are typically designed for rigid body motion, and thus struggle if applied to fluid motion estimation directly. Further, optical flow methods only focus on two consecutive frames without utilising historical temporal information, while the fluid motion (velocity field) can be considered a continuous trajectory constrained by time-dependent partial differential equations (PDEs). This discrepancy has the potential to induce physically inconsistent estimations. Here we propose an unsupervised learning based prediction-correction scheme for fluid flow estimation. An estimate is first given by a PDE-constrained optical flow predictor, which is then refined by a physical based corrector. The proposed approach outperforms optical flow methods and shows competitive results compared to existing supervised learning based methods on a benchmark dataset. Furthermore, the proposed approach can generalize to complex real-world fluid scenarios where ground truth information is effectively unknowable. Finally, experiments demonstrate that the physical corrector can refine flow estimates by mimicking the operator splitting method commonly utilised in fluid dynamical simulation.

Complex Locomotion Skill Learning via Differentiable Physics

Differentiable physics enables efficient gradient-based optimizations of neural network (NN) controllers. However, existing work typically only delivers NN controllers with limited capability and generalizability. We present a practical learning framework that outputs unified NN controllers capable of tasks with significantly improved complexity and diversity. To systematically improve training robustness and efficiency, we investigated a suite of improvements over the baseline approach, including periodic activation functions, and tailored loss functions. In addition, we find our adoption of batching and an Adam optimizer effective in training complex locomotion tasks. We evaluate our framework on differentiable mass-spring and material point method (MPM) simulations, with challenging locomotion tasks and multiple robot designs. Experiments show that our learning framework, based on differentiable physics, delivers better results than reinforcement learning and converges much faster. We demonstrate that users can interactively control soft robot locomotion and switch among multiple goals with specified velocity, height, and direction instructions using a unified NN controller trained in our system.

M2N: Mesh Movement Networks for PDE Solvers

Mainstream numerical Partial Differential Equation (PDE) solvers require discretizing the physical domain using a mesh. Mesh movement methods aim to improve the accuracy of the numerical solution by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, mesh movement methods, such as the Monge-Ampere method, require the solution of auxiliary equations, which can be extremely expensive especially when the mesh is adapted frequently. In this paper, we propose to our best knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampere method, our approach can greatly accelerate the mesh adaptation process, whilst achieving comparable numerical error reduction.

Aesthetic Photo Collage with Deep Reinforcement Learning

Photo collage aims to automatically arrange multiple photos on a given canvas with high aesthetic quality. Existing methods are based mainly on handcrafted feature optimization, which cannot adequately capture high-level human aesthetic senses. Deep learning provides a promising way, but owing to the complexity of collage and lack of training data, a solution has yet to be found. In this paper, we propose a novel pipeline for automatic generation of aspect ratio specified collage and the reinforcement learning technique is introduced in collage for the first time. Inspired by manual collages, we model the collage generation as sequential decision process to adjust spatial positions, orientation angles, placement order and the global layout. To instruct the agent to improve both the overall layout and local details, the reward function is specially designed for collage, considering subjective and objective factors. To overcome the lack of training data, we pretrain our deep aesthetic network on a large scale image aesthetic dataset (CPC) for general aesthetic feature extraction and propose an attention fusion module for structural collage feature representation. We test our model against competing methods on two movie datasets and our results outperform others in aesthetic quality evaluation. Further user study is also conducted to demonstrate the effectiveness.

Scalable Projection-Free Optimization

As a projection-free algorithm, Frank-Wolfe (FW) method, also known as conditional gradient, has recently received considerable attention in the machine learning community. In this dissertation, we study several topics on the FW variants for scalable projection-free optimization. We first propose 1-SFW, the first projection-free method that requires only one sample per iteration to update the optimization variable and yet achieves the best known complexity bounds for convex, non-convex, and monotone DR-submodular settings. Then we move forward to the distributed setting, and develop Quantized Frank-Wolfe (QFW), a general communication-efficient distributed FW framework for both convex and non-convex objective functions. We study the performance of QFW in two widely recognized settings: 1) stochastic optimization and 2) finite-sum optimization. Finally, we propose Black-Box Continuous Greedy, a derivative-free and projection-free algorithm, that maximizes a monotone continuous DR-submodular function over a bounded convex body in Euclidean space.

Unsupervised Learning of Particle Image Velocimetry

Particle Image Velocimetry (PIV) is a classical flow estimation problem which is widely considered and utilised, especially as a diagnostic tool in experimental fluid dynamics and the remote sensing of environmental flows. Recently, the development of deep learning based methods has inspired new approaches to tackle the PIV problem. These supervised learning based methods are driven by large volumes of data with ground truth training information. However, it is difficult to collect reliable ground truth data in large-scale, real-world scenarios. Although synthetic datasets can be used as alternatives, the gap between the training set-ups and real-world scenarios limits applicability. We present here what we believe to be the first work which takes an unsupervised learning based approach to tackle PIV problems. The proposed approach is inspired by classic optical flow methods. Instead of using ground truth data, we make use of photometric loss between two consecutive image frames, consistency loss in bidirectional flow estimates and spatial smoothness loss to construct the total unsupervised loss function. The approach shows significant potential and advantages for fluid flow estimation. Results presented here demonstrate that our method outputs competitive results compared with classical PIV methods as well as supervised learning based methods for a broad PIV dataset, and even outperforms these existing approaches in some difficult flow cases. Codes and trained models are available at https://github.com/erizmr/UnLiteFlowNet-PIV.

More Data Can Expand the Generalization Gap Between Adversarially Robust and Standard Models

Despite remarkable success in practice, modern machine learning models have been found to be susceptible to adversarial attacks that make human-imperceptible perturbations to the data, but result in serious and potentially dangerous prediction errors. To address this issue, practitioners often use adversarial training to learn models that are robust against such attacks at the cost of weaker generalization accuracy on unperturbed test sets. The conventional wisdom is that more training data should shrink the generalization gap between adversarially-trained models and standard models. However, we study the training of robust classifiers for both Gaussian and Bernoulli models under $\ell_\infty$ attacks, and we prove that more data may actually increase this gap. Furthermore, our theoretical results identify if and when additional data will finally begin to shrink the gap. Lastly, we experimentally demonstrate that our results also hold for linear regression models, which may indicate that this phenomenon occurs more broadly.