Title: Neural Green’s Functions
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neural Operators
Abstract: We introduce Neural Green’s Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green’s functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green’s Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green’s Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green’s Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9% across five shape categories, while
being up to 350 times faster than a numerical solver that requires computationally expensive meshing.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper offers a neural network based approach to solving Poissons equation with an approach that leverages the computational form of classical linear algebra solution. The main insight is the Shur inverse, the inverse of the interior of the Laplacian, analogous to the discrete Green's function, is approximated with a neural network. This is only possible because of the linearity of the Poisson problem.\nFollowing this the authors make the key insight that the Greens function is only geometry and not boundary value or source term dependent. Given this insight the problem is reduced to predicting Eigen-value scaled Eigen-vectors, a component from the decomposition of the Greens function.\n\nTo ensure the method is fully mesh independent this predicted quantity is also used to predict M and KLS.T (the surface to boundary sub-matrix of the Laplacian). \n\nExperiments are performed on 5 complex, non-convex geometries with Neural Green beating Transovler, the next best method, by >10% error reduction, with errors against an FEM approach in the 16-34% range but with a 350 times speed up.\n\nSTRENGTHS AND WEAKNESSES:\nFormulating the solution operator closely to the linear alegra computational structure is a strong mathematically grounded approach. Allowing to use a stronger inductive bias, utilising the linearity of Poissons equation and the stencil of the numerical method, replacing the computationally complex parts (matrix decomposition) with NN acceleration.\n\nThe work is clear, well presented and self contained. The benchmark datasets of the Poisson solves on the MCB dataset is a valuable contribution to the community.\n\nHowever I have some concerns to the value of the work. A 13.9% improvement is not huge, especially since the errors against an FEM approach still in the 16-34% range. So it's a 1-2% actual error improvement. Moreover the Transolver baseline can be applied to PDEs other than the linear Poisson equation.\n\nRuntime comparison - the paper compares against FEM runtime (which is good) but don't include the runtime of the other ML baselines.\nThere might be be a sitution where Transolver has a 22.1% error against FEM but a x1000 speedup versus Neural Greens that has a 18.9% error but a x330 speedup.\n\nThe authors critise the transformer style baselines saying \"but they require large-scale training datasets to achieve generalization.\" \nbut I couldn't see any analysis that they overcome the need to for large datasets or any evidence that there method generalizes better than the baselines beyond the 200/20 train/test split.\n\nAlthough the method is novel, the paper is clear, well explained, my prevailing concerns about the benefits compared to existing neural solvers leave my rating at weak reject unless the authors can convince me on the evaluation and usefulness of this work.\n\nQUESTIONS:\nWhat kind of neural network is used to predict $\\Phi$? In the theme of adding \"stronger priors\", would it be possible to make a stronger inductive bias for example roto-translational equvariance in this network?\n\nDid the authors try with any exotic feature engineering, ie signed distance function from geometry instead of positional coordinate?\n\nI understand equation (7) inspires the dot products between features, but in (6) G involves the inverse of K.L.K^T. Then how is it justified to use non-verted K/K^T multiplication?\n\nIf I could share for idea for the authors to be able to leverage their Poisson specialist method to other PDEs is taking an approach similar to\nZeng et al - \"PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems\", where the Laplace Block calves out the Laplacian solve, deferring other dynamics contributions from parallel NN architectures, GNN (or a NERF if requiring mesh free property).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce an operator learning framework - Neural Green's function. The idea is to embed Green's functions in the learning procedure for operators to enable data-efficient learning and deal with the issue of poor generalization in unseen domains, boundary functions and source terms. The authors propose approximating the Green’s function for a given problem domain allowing to compute solutions for arbitrary source and boundary functions by numerically evaluating the integral. They demonstrate superior generalization and speed-ups compared to other state-of-the-art approaches on the MCB dataset for Poisson equation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1) Approach: The central idea of incorporating Green’s functions into the framework design is quite interesting. Neural green's function is agnostic to source and boundary functions used for training by design and seems to generalize well across diverse geometries. \n2) Empirical validation for the Poisson equation: For the Poisson equation, the baseline comparisons with other state-of-the-art architectures and approaches, different source and boundary terms, and different irregular shapes are strong. \n3) Writing: The paper is well-written, and is relatively easy to follow. \n\nWeaknesses:\n1) Reproducibility: The code and data are unavailable. The details mentioned in the paper are certainly not enough to reproduce the results. For instance, the architectural details (number of layers, width) are missing, along with many other parameters such as number of interior and boundary points, activation functions, etc. \n2) Lack of examples: The study is restricted to a single PDE - Poisson equation. While the Poisson equation is explored in depth, an additional PDE like the Helmholtz equation (for instance, from reference [30] in the paper) would strengthen the results further. \n3) Limitations: The limitations of the proposed approach are not mentioned at all. I believe it is quite important to mention the limitations and key directions for future work that the authors feel are important. \n4) Weak baselines: the FEM solver is \"off the shelf\" (l.222), so not necessarily state of the art. Comparing the inference time of a (trained) neural operator in specific examples to an arbitary (i.e., not specifically designed) FEM solver is not reasonable. Poisson's equation is very well known in the scientific community and can be solved extremely efficiently (e.g. using multigrid). An off-the-shelf FEM solver probably cannot compare to a solver designed for this problem. It is ok to use it to generate the training data, but claiming superior performance in terms of inference time is not reasonable (but a very common mistake in physics-informed ML).\n\nQUESTIONS:\n1) Figures: I assume the authors have used the same color-bar for all the error plots in Figure 1. A color-bar for the true solution (column 2) and errors (columns 3-6) would be helpful to the reader. The same applies to Figures 3 and 4 in the main text and Figure 6 in Appendix D. \n2) Lines 41-42: I do not think it is quite true that neural PDE solvers generally match the accuracy of classical numerical methods. Optimization difficulties can often make it very hard for the neural PDE solvers to reach the accuracy of numerical solvers. I think this sentence needs to be rephrased. Could you elaborate your view? \n3) The authors compare the FEM run-time with the Neural Green's function evaluation/inference time, not the training time. I think it is perfectly fine to do so to stress the advantage during inference. However, it is also important to mention the training times and the fact that the major computational load for neural operators is offloaded to the training phase. Hence, the speed-ups during inference do not come without a price. Moreover, the operator frameworks are trained on data and thus have an advantage that they have more information, compared to solving a PDE for any particular instance with an FEM solver, which does not leverage any data from other simulations.  Could you elaborate your view and why you do not mention the training times?\n\nLIMITATIONS:\nNot mentioned at all, no separate limitation section.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Neural Green's Function (NGF), a mesh-free neural operator for linear Poisson PDEs with Dirichlet boundary conditions. NGF predicts a low-rank approximation of the Green's kernel through spectral feature learning, uses geometry-only conditioning for inference independence, and employs auxiliary MLP heads for solution evaluation. Tested on 3,400 heat-diffusion problems, NGF achieves 14% lower error than Transformer baselines and 350× speedup over FEM.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:** Unlike prior neural-operator works that learn a black-box mapping, their method explicitly reconstructs the Green’s kernel from geometry alone and exploits linear superposition. Demonstrates significant speed improvements compared to TetWild + Cholesky method. The paper is well organized as well. \n\n**Weaknesses:** Currently limited to Poisson + Dirichlet cases. Framework assumes single connected components. No discussion of stability/positive-definiteness. Interior-point sampling density is fixed (unspecified).\n\nQUESTIONS:\n1. What value of feature dimension d is used? How sensitive are results to it?\n\n2. Does the model ever predict negative or near-zero eigenvalues causing numerical issues?\n\n3. How are interior query points chosen during training and inference? Uniform sampling? Poisson-disk?\n\n4. Could NGF be combined with reduced-order models (e.g., POD) to further cut costs?\n\n5. Are there cases where NGF underperforms (e.g., extremely thin shells)? A qualitative failure example would be instructive.\n\n6. How difficult would extension be to Neumann/mixed boundary conditions or other linear elliptic operators (bi-Laplacian, anisotropic Laplacian)? \n\n7. For CAD models containing multiple disjoint solids, would spectral factorization remain stable?\n\n8. Since $G_{\\theta}$ is built from learned features, how do authors guarantee $KLK^{\\top}$ invertibility and positive mass prediction (beyond Soft-plus non-negativity)?\n\n9. Would performance degrade when evaluating on denser grids than training grids?\n\n---\n\nOverall, This paper presents a reasonable operator-learning framework that applies classical Green’s function analysis to solve Poisson’s equations. While the method is clearly explained and the results are promising on the presented benchmarks, the overall contribution is incremental rather than groundbreaking. Some aspects, such as the limitations to specific PDE types, lack of uncertainty quantification, and limited discussion of generalizability, prevent a higher recommendation. With further empirical validation and broader evaluation, the impact and significance of the work could be strengthened. At its current stage, I consider the contribution modest but relevant for the NeurIPS community.\n\nLIMITATIONS:\nThe paper does not explicitly articulate its limitations, but the proposed method is confined to specific types of partial differential equations (PDEs) and lacks uncertainty quantification, offering only limited discussion on generalizability.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a method that learns to predict neural approximate Green's functions for Poisson's equation. The approximate Green's function is a low-rank approximation formed from a learned domain-dependent pointwise feature map. The predicted Green's function yields an approximate solution operator that generalizes under changes in domain shape, boundary conditions, and source function.\n\nSTRENGTHS AND WEAKNESSES:\nThe results seem pretty impressive. However, the evaluation is on only a limited number of shapes (test set of 20). Since the main point of the paper is generalization to arbitrary shapes, it might be good to stress-test the method on more shapes (e.g., from a completely different collection or category). I am also not sure what you mean in composing combinations of domain, boundary function, and source function. E.g., a given boundary function is defined on the boundary of a particular domain. What does it mean to test the same boundary function across two different domains?\n\nIt might be good to evaluate the results quantitatively using the H^1 metric, rather than just L^2 as in equation (12). Given a source function $f$ in $L^2$, the solution to Poisson's equation should lie at least in H^1 in the interior, and it would be good to ensure that not only do the pointwise values of your approximate solution match the ground truth well, but also that their derivatives match well.\n\nTo give contex to your runtime comparison in Table 2, it would be fair to note also the total time used for training. (Though I understand the point that *at test time* for an individual domain, the neural proxy is much faster.)\n\nQUESTIONS:\n- What do you mean by \"leveraging the eigendecomposition of Green's functions [by] reconstructing them from features extracted from problem domains\" (111-112)? Are the features the eigenvalues/eigenfunctions? Or some approximation thereof?\n- Does your method generalize beyond Dirichlet boundary conditions?\n- How does your method compare to boundary element / boundary integral methods? Or Monte Carlo/Walk-on-spheres methods, which also support pointwise queries without requiring mesh generation or a global linear solve? When is it advantageous to learn a proxy solution operator vs., for example, using (possibly denoised) Monte Carlo estimates?\n- The method represents the geometry of the domain through the sampled query points. Thus the query points are doing double duty as points at which the Green's function and solution are evaluated, and as an input that conditions the predicted solution. How much does the learned solution depend on the details of this sampling? Ideally, the predicted solution at a fixed point $p$ in the domain should be invariant to the choice of other query points $Q$. Does the choice of query points matter at test time? Only during training? Or are the results nearly invariant to these choices, as one would hope?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}