Title: Boundary-Value PDEs Meet Higher-Order Differential Topology-aware GNNs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Partial differential equations

Neural operator

Higher-order graph neural network

Exterior differential calculus

Electromagneticism
Abstract: Recent advances in graph neural network (GNN)-based neural operators have demonstrated significant progress in solving partial differential equations (PDEs) by effectively representing computational meshes. However, most existing approaches overlook the intrinsic physical and topological meaning of higher-order elements in the mesh, which are closely tied to differential forms. In this paper, we propose a higher-order GNN framework that incorporates higher-order interactions based on discrete and finite element exterior calculus. The time-independent boundary value problems (BVPs) in electromagnetism are instantiated to illustrate the proposed framework. It can be easily generalized to other PDEs that admit differential form formulations. Moreover, the novel physics-informed loss terms, integrated form estimators, and theoretical support are derived correspondingly. Experiments show that our proposed method outperforms the existing neural operators by large margins on BVPs in electromagnetism. Our code is available at https://github.com/Supradax/Higher-Order-Differential-Topology-aware-GNN.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel method called DEC-HOGNN in the category of neural operators. It makes use of higher-order graph neural networks that can handle interactions between simplices of different orders, not just same orders. It also designs an encoder-decoder structure for differential forms of different orders involved to make predictions. By writing down fields as differential forms, the physical losses can also be easily implemented. This approach is tested on boundary value problems in electrostatics and magnetostatics for both 2D and 3D.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. This is an interesting attempt to implement physical losses elegantly and effectively in neural operators, which an inspiring and meaningful first exploration for AI for PDE community. This method does not require differentiation for physical losses.\n2. The mathematical motivations of this work are demonstrated in an easy to follow manner.\n3. There are clear proofs for important theorems mentioned in this paper, including one universal approximation property for this proposed neural operator.\n\nWeaknesses:\n1. Though a large part of this paper is devoted to FEEC, the data generation is still achieved with traditional FEM methods, but not FEEC. This point makes this work not in a \"pure\" FEEC manner.\n2. The datasets explored in the experiments of this work do not cover a wide enough range of problems. Only electrostatics and magnetostatics are covered, which does not provide enough evidence for the superiority of this approach.\n3. The inference speed is not discussed and compared with other methods.\n\nQUESTIONS:\n1. How are the boundary conditions $\\nabla \\phi = u(\\bf{x}), \\bf{x} \\in \\partial \\Omega$ implemented as an input for this neural operator?\n2. Related to point 1 in weaknesses, why are the data still generated with traditional FEM, instead of FEEC? If data are generated with FEEC, then no encoding is required and all differential forms are naturally solved out.\n\nLIMITATIONS:\nYes, the limitations are discussed in this paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed a novel GNN neural operator (NO) that is topology-aware by leveraging discrete exterior calculus (DEC). The advantage of this novel NO includes: 1. preserving the topological structure of the manifold which is also the problem domain of PDE; 2. It can turn some equations involving divergence and curl into boundary conditions by Stokes' theorem, such that the physical loss from equation no longer needs taking derivatives, e.g., divergence and curl, like PINN does.\n\nIn order to leverage the DEC, the authors designed a GNN that handles 4 types of neighborhood (see eq (8)-(12)). In experiments of 2D and 3D problems, the proposed method outperforms all baselines and showing strong advantage.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\nThis paper is highly innovative and origin by introducing DEC to learning operators with GNN. As mentioned in summary, there are two advantages: 1. Preserving topology structure of the manifold; 2. Simplifying the equation constraints to boundary constraints (see eq (19)-(22)). In prior works, topology structure is either ignored (e.g., transformer-based NO and FNO) or partially encoded (with graph adjacency of GNN). In contrast, DEC utilizes 4 types of neighborhood, which is a complete methodology. Ablation study (Table 2) shows that the complete setting outperforms incomplete ones. The second advantage naturally enables the proposed method to incorporate physics-informed loss without taking derivatives (for some equations, e.g., Maxwell, Navier-Stokes, etc.).\n\nThe paper is well written and elegantly presented. Also, for self-completeness, the paper provides universal approximation analysis.\n\n**Weakness**\n\n1. In Table 3, I think the bold numbers are the best of each column. Please clarify accordingly to avoid confusion.\n2. The current data is generated with finite element method, which may not be compatible with DEC (as the discussion in line 104-105). If data generation by DEC is not so convenient, this may limit the application of the method to some extent. \n3. Typo? On line 476, $H\\in\\Omega^1(\\star M)$ -> $ H\\in\\Omega^1(M)$ to align with Fig. 2.\n\nQUESTIONS:\nN.A.\n\nLIMITATIONS:\nN.A.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a neural operator framework for solving boundary value problems (BVPs) governed by partial differential equations (PDEs). The key innovation lies in leveraging Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) to explicitly encode higher-order topological structures (nodes, edges, faces, cells) as k-simplex features in graph neural networks (HOGNNs). This allows for principled modeling of scalar and vector fields as differential forms, enabling a topologically-aware neural solver.\n\nThe approach includes physics-informed encoder-decoder mappings between vector fields and forms, integrated physical loss functions that correspond to conservation laws (e.g., Gauss’s law), and theoretical guarantees of universal approximation for certain BVP classes. There are limited experimental evaluations on 2D and 3D electrostatics and magnetostatics.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths*\n\n- The methodology is technically sound and grounded in well-established mathematical frameworks (DEC and FEEC), with rigorous theoretical support (e.g., encoder-decoder preservation, universal approximation).\n- This work is an important step toward building structure-preserving, physically grounded neural solvers for BVPs, providing a new approach. The proposal to use higher-order GNNs aligned with exterior calculus opens a new direction for learning-based PDE solvers in domains like electromagnetism, fluid dynamics, and elasticity.\n- The synthesis of DEC/FEEC with HOGNNs for BVPs appears novel and nontrivial. While GNNs and physics-informed learning are well-trodden, the topological interpretation via differential forms and the dual-manifold message passing are particularly unique contributions (for solving PDEs).\n- The physics-informed loss design is very elegant and potentially efficient! I love how the work leverages integrated features (which appear naturally in DEC) directly, which avoids the need for expensive quadrature.\n\n*Weakness*\n- This is a very dense paper, with some parts written with excessive jargon (particularly early on), making it hard for a broader audience to parse. Key ideas (like Eq. 5) are underexplained at first reading. \n- Figures (e.g., Figure 1 and 2) are central to the understanding but require much more extensive captions and annotations to be fully self-contained.\n- It is unclear why the 3D magnetostatic results are not competitive (relative to other baselines) even though the model is designed to generalize naturally to 3D.\n-  The comparison misses modern BVP-specific baselines such as FNO, CNO, SCOT, etc., which are more appropriate than DeepONet or Transolver (designed for time-evolving systems).\n- The experimental results are very perfunctory, and should be strengthened. \n- While the stated focus is on time-dependant PDEs (in the abstract), all analysis and results are for the time-**independant** PDE. This is rather confusing.\n\nQUESTIONS:\n- The formulation of the neural operator in Eq. (5) is too abstract and hard to interpret until the reader reaches the concrete instantiation in Eq. (7). The authors could either move the motivating example earlier or introduce a simplified case (e.g., scalar Poisson) first.\n\n- Figures 1 and 2 are critical but lack self-contained descriptions. For a technical audience unfamiliar with DEC or the specific PDE being solved, these need extensive legends explaining what each symbol and arrow denotes.\n\n- More focussed experiments: Pick 2-3 shapes in 2D and 3D (exhibiting different topological characteristics), and comprehensively evaluate performance against other state of art. Additionally, while the paper briefly notes that performance is affected by mesh shape and quality, this is an important limitation for deployment in practical engineering settings. Additional empirical analysis (e.g., mesh convergence study) would strengthen the impact.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nsolve boundary value PDE problems. The authors incorporate discrete and finite element exterior calculus into GNNs to deal with geometrical relationships between differential forms (e.g., charge density and electric field). They also derived the physics-informed loss to measure the consistency of the output against the considered physical law. In addition, the authors provided the universal approximation theorem for Poisson problems, justifying the use of the model for electromagnetic phenomena. The experimental results suggest that the proposed method has a high capacity to predict electromagnetic phenomena for 2D and 3D settings.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The writing is well-construcgted and easy to follow. The authors reveal the connection of the present method to existing ones from the viewpoint of DEC (discrete exterior calculus), clarifying the position of the research.\n* The derivation of physics-informed loss for GNN is useful. Thanks to the DEC formulation, a certain physical constraint turned out to be the sum of integration, which is easy to compute.\n* The method is demonstrated to have high accuracy compared to the considered baseline models, showing the expressibility of the model.\n\nWeakness:\n* The relation to existing exterior-calculus-based or simplicial-complex-based methods is not clearly stated. For instance, differences and advantages against research on cellular complexes [Alain+ ICML 2024] and simplicial complexes [Ebli+ NeurIPS Workshop 2020] could be stated, and possibly, compared experimentally.\n* There is no evaluation of computation time. Since the classical solvers have high accuracy (with possibly a lot of computation resources), the evaluation of the accuracy alone is not enough. Therefore, the reviewer recommends evaluating the speed-accuracy tradeoff with changing spatiotemporal resolution (at least for classical solvers).\n\nMinor points:\n* Simplex complex (l.63 p. 2) could be called simplicial complex instead.\n* Since x is not used in the PDE, $x \\in \\Omega$ in Equation 6 is confusing. Either using x in the PDE or writing $\\mathrm{in} \\ \\Omega$ would be preferable.\n\nQUESTIONS:\n* In the experiments, how much was the error of PDE constraints, e.g., divergence-free condition?\n* What would be necessary to extend the method for other type of PDEs, e.g., Navier–Stokes equations?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}