Computational Intelligence (CI) techniques have shown great potential as a surrogate model of expensive physics simulation, with demonstrated ability to make fast predictions, albeit at the expense of accuracy in some cases. For many scientific and engineering problems involving geometrical design, it is desirable for the surrogate models to precisely describe the change in geometry and predict the consequences. In that context, we develop graph neural networks (GNNs) as fast surrogate models for physics simulation, which allow us to directly train the models on 2/3D geometry designs that are represented by an unstructured mesh or point cloud, without the need for any explicit or hand-crafted parameterization. We utilize an encoder-processor-decoder-type architecture which can flexibly make prediction at both node level and graph level. The performance of our proposed GNN-based surrogate model is demonstrated on 2 example applications: feature designs in the domain of additive engineering and airfoil design in the domain of aerodynamics. The models show good accuracy in their predictions on a separate set of test geometries after training, with almost instant prediction speeds, as compared to O(hour) for the high-fidelity simulations required otherwise.
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ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
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Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations
Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. The proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.
Backpropagation algorithm has been widely used as a mainstream learning procedure for neural networks in the past decade, and has played a significant role in the development of deep learning. However, there exist some limitations associated with this algorithm, such as getting stuck in local minima and experiencing vanishing/exploding gradients, which have led to questions about its biological plausibility. To address these limitations, alternative algorithms to backpropagation have been preliminarily explored, with the Forward-Forward (FF) algorithm being one of the most well-known. In this paper we propose a new learning framework for neural networks, namely Cascaded Forward (CaFo) algorithm, which does not rely on BP optimization as that in FF. Unlike FF, our framework directly outputs label distributions at each cascaded block, which does not require generation of additional negative samples and thus leads to a more efficient process at both training and testing. Moreover, in our framework each block can be trained independently, so it can be easily deployed into parallel acceleration systems. The proposed method is evaluated on four public image classification benchmarks, and the experimental results illustrate significant improvement in prediction accuracy in comparison with the baseline.
Deep learning surrogate models are being increasingly used in accelerating scientific simulations as a replacement for costly conventional numerical techniques. However, their use remains a significant challenge when dealing with real-world complex examples. In this work, we demonstrate three types of neural network architectures for efficient learning of highly non-linear deformations of solid bodies. The first two architectures are based on the recently proposed CNN U-NET and MAgNET (graph U-NET) frameworks which have shown promising performance for learning on mesh-based data. The third architecture is Perceiver IO, a very recent architecture that belongs to the family of attention-based neural networks--a class that has revolutionised diverse engineering fields and is still unexplored in computational mechanics. We study and compare the performance of all three networks on two benchmark examples, and show their capabilities to accurately predict the non-linear mechanical responses of soft bodies.
To facilitate widespread adoption of automated engineering design techniques, existing methods must become more efficient and generalizable. In the field of topology optimization, this requires the coupling of modern optimization methods with solvers capable of handling arbitrary problems. In this work, a topology optimization method for general multiphysics problems is presented. We leverage a convolutional neural parameterization of a level set for a description of the geometry and use this in an unfitted finite element method that is differentiable with respect to the level set everywhere in the domain. We construct the parameter to objective map in such a way that the gradient can be computed entirely by automatic differentiation at roughly the cost of an objective function evaluation. The method produces optimized topologies that are similar in performance yet exhibit greater regularity than baseline approaches on standard benchmarks whilst having the ability to solve a more general class of problems, e.g., interface-coupled multiphysics.
Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models inadequately treat boundary conditions essential for the reliable prediction of such problems. In addition, because of the locally connected nature of GNNs, it is difficult to accurately predict the state after a long time, where interaction between vertices tends to be global. We present our approach termed physics-embedded neural networks that considers boundary conditions and predicts the state after a long time using an implicit method. It is built based on an E(n)-equivariant GNN, resulting in high generalization performance on various shapes. We demonstrate that our model learns flow phenomena in complex shapes and outperforms a well-optimized classical solver and a state-of-the-art machine learning model in speed-accuracy trade-off. Therefore, our model can be a useful standard for realizing reliable, fast, and accurate GNN-based PDE solvers. The code is available at //github.com/yellowshippo/penn-neurips2022.
Solving complex fluid-structure interaction (FSI) problems, which are described by nonlinear partial differential equations, is crucial in various scientific and engineering applications. Traditional computational fluid dynamics based solvers are inadequate to handle the increasing demand for large-scale and long-period simulations. The ever-increasing availability of data and rapid advancement in deep learning (DL) have opened new avenues to tackle these challenges through data-enabled modeling. The seamless integration of DL and classic numerical techniques through the differentiable programming framework can significantly improve data-driven modeling performance. In this study, we propose a differentiable hybrid neural modeling framework for efficient simulation of FSI problems, where the numerically discretized FSI physics based on the immersed boundary method is seamlessly integrated with sequential neural networks using differentiable programming. All modules are programmed in JAX, where automatic differentiation enables gradient back-propagation over the entire model rollout trajectory, allowing the hybrid neural FSI model to be trained as a whole in an end-to-end, sequence-to-sequence manner. Through several FSI benchmark cases, we demonstrate the merit and capability of the proposed method in modeling FSI dynamics for both rigid and flexible bodies. The proposed model has also demonstrated its superiority over baseline purely data-driven neural models, weakly-coupled hybrid neural models, and purely numerical FSI solvers in terms of accuracy, robustness, and generalizability.
We consider the approximation of a class of dynamic partial differential equations (PDE) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation. Our analyses show that, with feed-forward neural networks having two hidden layers and the $\tanh$ activation function, the PINN approximation errors for the solution field, its time derivative and its gradient field can be effectively bounded by the training loss and the number of training data points (quadrature points). Our analyses further suggest new forms for the training loss function, which contain certain residuals that are crucial to the error estimate but would be absent from the canonical PINN loss formulation. Adopting these new forms for the loss function leads to a variant PINN algorithm. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture the solution well.
Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.
Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias and developed geometrically equivariant Graph Neural Networks (GNNs) to better characterize the geometry and topology of geometric graphs. Despite fruitful achievements, it still lacks a survey to depict how equivariant GNNs are progressed, which in turn hinders the further development of equivariant GNNs. To this end, based on the necessary but concise mathematical preliminaries, we analyze and classify existing methods into three groups regarding how the message passing and aggregation in GNNs are represented. We also summarize the benchmarks as well as the related datasets to facilitate later researches for methodology development and experimental evaluation. The prospect for future potential directions is also provided.
How can we estimate the importance of nodes in a knowledge graph (KG)? A KG is a multi-relational graph that has proven valuable for many tasks including question answering and semantic search. In this paper, we present GENI, a method for tackling the problem of estimating node importance in KGs, which enables several downstream applications such as item recommendation and resource allocation. While a number of approaches have been developed to address this problem for general graphs, they do not fully utilize information available in KGs, or lack flexibility needed to model complex relationship between entities and their importance. To address these limitations, we explore supervised machine learning algorithms. In particular, building upon recent advancement of graph neural networks (GNNs), we develop GENI, a GNN-based method designed to deal with distinctive challenges involved with predicting node importance in KGs. Our method performs an aggregation of importance scores instead of aggregating node embeddings via predicate-aware attention mechanism and flexible centrality adjustment. In our evaluation of GENI and existing methods on predicting node importance in real-world KGs with different characteristics, GENI achieves 5-17% higher NDCG@100 than the state of the art.
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