Structure-Preserving Operator Learning
Nacime Bouziani, Nicolas Boull\'e
TL;DR
This paper introduces structure-preserving operator networks (SPONs) that leverage finite element discretizations to accurately learn operators from PDE-driven systems, preserving key physical properties and handling complex geometries.
Contribution
The paper proposes a novel family of SPON architectures that preserve continuous system properties at the discrete level and operate efficiently on complex geometries, with theoretical guarantees.
Findings
SPONs can exactly enforce boundary conditions.
SPONs handle complex geometries effectively.
Multigrid-inspired SPONs improve performance and efficiency.
Abstract
Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that…
Peer Reviews
Decision·ICLR 2025 Conference Withdrawn Submission
1. The authors introduced the FEM framework into neural PDE solvers, which generalized previous works that formulate the problem as a discrete mapping. This formulation is well-suited for problems on irregular domains. Notably, this formulation enables discretization-invariance in GNN architectures. 2. The proposed method is enhanced with multigrid methods, which improves the performance and efficiency of the network.
1. Lack of comparison with previous GNN-based solvers, both methodologically and experimentally. Based on my understanding, SPON (without -MG) differs from GNN solvers only on its encoder and decoder, i.e., a different input/output embedding scheme. It is not clear whether the proposed method still brings improvement, in terms of both accuracy and efficiency, over these GNNs. 2. Soundness of comparison. The authors compared their model to FNOs on 2D Poisson equation with Dirichlet BC, and repor
1. The paper is well-written and well-organized, making it easy to understand. 2. The idea is novel. It addresses significant issues relevant to the AI4PDE field, highlighting its importance and potential applications. 3. The proposed method can automatically satisfy the boundary conditions. 4. The author provides several theoretical insights.
The main issue with this paper lies in its experimental design. I listed my major concerns as follows: 1. The authors conducted only two experiments, one on the Poisson equation and one on the flow around a cylinder. This limited experiments may affect the generalizability and reliability of the proposed methods. 2. For the Poisson equation experiment, the authors only compared their method with basic approaches (like FNO and DeepONet), which are three years old, without including more advance
This paper provides a combination of operator learning with FEM and with multigrid methods. It is clearly written and easy to follow.
The paper suffers from multiple weaknesses: a) There is no clear comparison to other methods. For example, the authors only compare against FNOs and DeepONets. DeepONets are not state-of-the-art and FNOs are not Representation Equivalent Neural Operator, as in Bartolucci et. al. that the authors cite. Therefore, in my understanding, the authors only compare for a very easy example, to methods that are bound to fail from the start, so comparison is not proper. I believe that the authors should c
1) The authors developed a new operator learning framework that maintains structure-preserving properties derived from finite element methods. 2) They established approximation bounds for a broad class of operators. 3) They conducted experiments on various PDEs, demonstrating the effectiveness of their approach. 4) They empirically demonstrated a trade-off between accuracy and efficiency. 5) The software is open-source and easy to use. 6) The paper is well-written, technically sound, and easy to
1) There is a lack of baselines across all the experiments you conducted. For the Poisson equation, where the grid is uniform, one of the convolution-based architectures [1][2][3] could have been easily tested. In the other experiments, no baselines were included, despite the availability of numerous models that handle nonuniform discretizations [4][5]. While I understand that your goal isn't necessarily to outperform existing models, it would still be valuable for the community to see how your
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Taxonomy
TopicsAdvanced Measurement and Metrology Techniques · Control Systems in Engineering · Robotic Mechanisms and Dynamics