Title: O-MMGP: Optimal Mesh Morphing Gaussian Process Regression for Solving PDEs with non-Parametric Geometric Variations
Venue: NeurIPS 2025
Accepted: no
Keywords: Gaussian process

Mesh morphing

Reduced order modeling
Abstract: We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.


Peer reviews:
comments: SUMMARY:
The authors used an optimization process on Gaussian Processes, used as surrogates for the true functions that map parameters and "PODs" to "optimal PODs", to find the optimal configuration (parameters) for a complex physical problem, provided in the form of a PDE.

STRENGTHS AND WEAKNESSES:
# Strengths
- The problem seems highly relevant and the example use case is well chosen
- While the required preliminaries are very advanced, the authors did ok in explaining the material in an understandable fashion. Still, there is room to provide intuitions and refer to the application areas throughout the discussion, making it even more understandable.

# Weaknesses
- **The authors did not keep to the nine pages limit. Gaining a (slight) unfair advantage over the remaining submissions.**

- Figure 1 is unclear, I can't see the second airfoil without zooming in to 200%

## Minor
- Positioning of the plots in Figure 2
### Typos
- 14: varibale
- 117: depends
- 148: provides, finds
- 152: maximize
- 182: lies
- 250: "proves to be much efficient numerically"
- Legend in Figure 5 b)
- 282: decays

QUESTIONS:
- What is the difference between $\phi^\text{geo}$ and $\phi^\text{opt}$?
- Why don't you combine the GPs $\mathcal{R}$ and $\mathcal{O}$ into a single multi-output GP that learns $\gamma$ and $\beta$ together?
- Could you give an estimate on how hard/easy it is to find the mappings $\phi$ for various different problems? i.e. how easily applicable is your approach to different problems. Can it be done algorithmically or are all preparatory calculations done manually?

LIMITATIONS:
- As mentioned in the questions, the difficulty of applying the authors method was not discussed
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-24

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents O-MMGP, a method for solving PDEs where the solution domains vary non-parametrically. The approach first morphs all domains to a common reference using RBF interpolation. Then, a mapping function is optimized to align solution fields such that energy is maximized in the first few POD modes. Once the alignment is done, POD is performed and two GPR models are trained: one to predict the mapping coefficients, and another to predict the reduced solution coefficients. At inference time, the method applies the predicted mappings and reconstructs the full solution. The method is tested on a synthetic advection-reaction example and on the AirfRANS dataset from ML4CFD, where it shows improved MSE over both MMGP and a manually aligned baseline.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n- The method improves the alignment of solution fields, helping make low-rank approximation more effective—this is supported by the eigenvalue plots. It keeps the original MMGP structure intact, adding just one step offline while keeping online inference efficient. \n- The optimization for alignment is explained clearly, with enough detail to follow. Using a coarse mesh for alignment and interpolating to a fine mesh is a simple but useful way to reduce cost.\n- The geometric alignment could possibly transfer beyond the intended application, but this was not demonstrated.\n\n\n### **Weaknesses**\n- The scope is too limited: it is only relevant to those who are dealing with parametric PDEs with geometric variations AND engaging POD AND with the objective to compress ensemble from multiple geometry for training a surrogate like GPR. The framework cannot be applied to new end-to-end AI methods. It seems to me a niche application, not the groundbreaking work expected for NIPS papers.\n- Do the PODs change based on initial conditions? How do we handle this? It seems the POD approach runs into many different challenges.\n- The alignment step is based on a non-convex optimization over functions, but the paper doesn't provide any analysis of how reliably this process works. There’s no mention of how often the mappings remain valid (bijective) or whether they break down—like folding or other distortions—in practice.\n- The method assumes that all domains have the same topology and that boundary control points can be matched across samples. This is a big ask in real-world problems, where clean correspondences are rare. - The paper doesn’t test or discuss how the method holds up when these assumptions don’t quite hold—like when control points are noisy or misaligned.\n- There are several hyperparameters that clearly matter (number of modes, penalty weights, step sizes, etc.), but no ablation or sensitivity study is included. This makes it hard for others to know how to tune the method or trust that it will work out-of-the-box on different problems.\n- The experiments are narrow: one toy problem and one 2D real-world dataset. There’s no test on 3D problems, other types of PDEs, or geometries with more extreme variation. Without that, it's hard to know if the method really works beyond the settings chosen here.\n- The comparisons are also limited. The paper only compares against MMGP and a manually aligned baseline. There’s no comparison to modern operator-learning models like DeepONet or Geo-FNO, which could potentially be paired with the same alignment step. Without that, the improvement over existing methods feels isolated and hard to evaluate.\n- Even though the method uses GPR, which provides predictive uncertainty, the paper doesn’t actually report or analyze any of it. There’s no look at calibration, error bars, or how uncertainty behaves in test or OOD settings. That’s a missed opportunity for a UQ-related contribution.\n- Runtime is only given for a single case, and there’s no discussion of how cost scales with mesh size, number of samples, or problem dimension. That makes it difficult to understand how practical the method would be in larger or more realistic settings.\n- The results are shown without any variability analysis—just single-run metrics with no error bars. It’s unclear how consistent or significant the improvements are, especially when the gains are small. This weakens confidence in the overall empirical contribution.\n\nQUESTIONS:\nJust to note before the questions—some of the earlier concerns feel pretty central. I hope the authors take a moment to address those too.\n- Can you explain how runtime and memory usage scale with bigger meshes, higher dimensions, or more training samples?\n- Did you track how often the mapping step actually produces valid, bijective results? Were there any cases where the mapping folded or failed during training?\n- Have you tried adding noise to the control points or tweaking the boundary shapes a bit? How much does that affect the mapping and final predictions?\n- How sensitive is the method to things like the number of POD modes or the penalty and diffusion terms in the optimization? Any tips or heuristics for picking those values?\n- The paper doesn’t test training on a coarse mesh and then running on a finer mesh to see if accuracy holds. Adding a simple “train coarse, test fine” experiment or discussing how interpolation might affect results would make the approach more trustworthy.\n- Have you tested this approach on other types of PDEs or in 3D settings? If not, what kind of issues do you expect to run into?\n- Why didn’t you compare to operator-learning methods like Geo-FNO or DeepONet applied to the aligned fields? Do you think those models might outperform your GPR approach?\n- Does the uncertainty from GPR actually match the prediction error? For example, do variances go up when you move away from the training data?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe author proposes an optimal mesh morphing Gaussian process (O-MMGP) to achieve the non-linear dimensionality reduction for efficient and accurate modeling of the parametric PDEs.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. The author combines optimization with the MMGP to gain improvements over the cutting-edge methods.\n2. The author conducts experiments in NeurIPS ML4CFD competition and claims a better result, compared to the wining solution.\n3. The author gives thorough proofs for solving the optimization.\n\nWeakness:\nIn general, the paper is not well-written. Some important concepts are not well explained. The methodology is hard to understand. Please see my following questions in the Question section.\n\nQUESTIONS:\n1.\tSome concepts are not explained well. For example, what is Kolmogorov width and why does the decaying of the width cause trouble? What is geometric variation? What is the first-principle mode?  What is MMGP method?\n2.\tThe author should also give references when introducing these concepts. \n3.\tIn the literature review 1.3, the author doesn’t mention the drawbacks of current learning-based methods to solve the problem. Hence, their own design is not well-motivated. \nI am not an expert in this domain, and the methodology is hard for me to understand. In general, I have the following questions about the technical details. \n4.\tWhat is the functional in Eq. (3) and Eq. (4)? What does this objective mean and why should we introduce them?\n5.\tIn Section 3.1.2, why should we use the Gaussian process regression to estimate the mapping? Did the author test other regression methods?\n6.\tSection 4 is very short. I doubt that it can be treated as an independent Section. What’s the relation between Sections 3, 4, and 5? \n7.\tIs solving the Optimization in Eq. (3) (Section 5) your unique contribution? What’s the existing work for this optimization? What are their drawbacks? What’s the novelty for the author’s algorithm? \n8.\tIt seems that the author’s method is based on the so-called mesh morphing Gaussian process (MMGP). However, MMGP is only introduced in the Experiment part (Section 6.4). I suggest that the author to move this part to the Preliminary and Methodology, explain more about the difference and the development from MMGP to O-MMGP. \n9.\tFinally, the author doesn’t mention about the potential drawbacks and the future work for their methods. \n10.\tThe improvements of the proposed method to the state-of-the-art is not significant.\n\nLIMITATIONS:\nThe authors need to elaborate on the limitations and future work for their proposed framework. This part is not present in the current manuscript.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 1, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces O-MMGP, a framework that combines mesh morphing, non-linear dimensionality reduction, and Gaussian Process Regression to address the challenge of efficiently solving parametric PDEs with significant geometric variations and non-reducible problems (e.g., those featuring shocks and moving discontinuities). By defining and optimizing so-called “optimal mappings,” the approach enables a more compact representation of the solution fields, facilitating efficient regression and approximation. The methodology automates alignment and reduces reliance on manual feature tracking, and is empirically validated on both a synthetic advection-reaction PDE and the ML4CFD NeurIPS 2024 airfoil benchmark, with comparison to state-of-the-art and competition-winning methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. Technical Strength and Theoretical Foundation: The formulation of the optimal mapping problem is mathematically well-grounded, providing a rigorous optimization framework for enhancing the compressibility of solution fields with slowly decaying Kolmogorov width. The use of an explicitly derived functional and an infinite-dimensional gradient algorithm shows solid theoretical backing.\n2. Meaningful Advancement over State-of-the-Art: The approach enables automatic, principled multi-modal alignment for non-parametric geometry PDEs, which traditionally suffer from high-dimensional irreducibility when shocks or discontinuities move. \n3. Empirical Rigor: The paper conducts comprehensive experiments on both a synthetic advection-reaction PDE (Figure 2) and the widely adopted AirfRANS/ML4CFD dataset (Section 6.2), comparing the proposed O-MMGP to both original MMGP and the NeurIPS 2024 competition winner.\n\nWeakness:\n1. Experimental Scope and Diversity: The experimental validation, while strong in its target domains, focuses primarily on two examples: a synthetic 2D PDE and the ML4CFD/Gaussian Process airfoil data. No evidence is provided for higher-dimensional problems (e.g., 3D shapes) or alternative domains, limiting claims of broad generalization.\n2. Baseline and Ablation Details: The paper compares O-MMGP against MMGP and one expert-tuned baseline (the NeurIPS winner), but misses some possible baselines, e.g., recent neural operator methods or advanced geometric deep learning surrogates for variable domains [such as Geo-FNO, DeepONet in more detail]. The ablation with respect to various components (optimal mapping, field transformation, regression model choice) could be fleshed out (Section 6.4 references parameter choices but does not deeply analyze their impact).\n3. Parameter Sensitivity and Hyperparameter Selection: While the paper specifies values for key parameters (e.g., c1, c2, r), it does not systematically analyze sensitivity or robustness to these choices. The effect of continuation parameters and the impact of the elastic penalty for mapping bijectivity are mentioned but not quantitatively explored (see Section 5.2, and empirical discussion).\n4. Scalability and Computational Cost: The method involves potentially expensive optimization in infinite-dimensional function spaces, with multiple iterative or continuation steps. While the cost is referred to (see Section 6.4, and appendix references), there is limited main-text quantification of run-time or resource needs for larger-scale or more complex problems, and the impact of mesh size or sample count is not discussed.\n\nQUESTIONS:\nScalability to Larger and 3D Problems: Could the authors provide quantitative insight or runtime scaling on how O-MMGP would perform for larger mesh sizes, 3D domains, or higher sample counts? Are there bottlenecks related to the mesh-morphing or the infinite-dimensional optimization that would limit real-world adoption?\n\nParameter Sensitivity and Robustness: What is the sensitivity of the approach to choices of c1, c2, r, and POD truncation? Would alternative choices substantially degrade performance, or is there a principled way to set them in new problems?\n\nAutomated Curved Boundary Handling: As future work is planned to generalize to curved boundaries, are there conceptual or computational obstacles the authors foresee? How critical is this limitation in practice for the domains the authors believe O-MMGP could address?\n\nUncertainty Quantification: GPR, in general, enables predictive uncertainty. Did the authors make any use of these uncertainty estimates in the downstream tasks or error analysis? If so, could they provide an empirical illustration, for example, on the out-of-distribution (OOD) scenarios?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper presents a novel nonlinear dimensionality reduction technique for solving parametric PDEs with non-parametrized geometric variations. Combining with morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression, this approach enables efficient and accurate modeling of complex parametric PDEs defined on variable geometries. The experimental results demonstrate the efficiency performance comparable to state-of-the-art methods, including the winning solution of the ML4CFD NeurIPS 2024 competition.

STRENGTHS AND WEAKNESSES:
Strengths:
By introducing nonlinear dimensionality reduction, the model contains the three approximations parts, which maintains the geometric transformation from reference configuration to the optimal configuration and couples with field-dependent optimization. This method combines two Gaussian process regression. 
The novelty of the algorithm lies in the fact that it automatically aligns the functions on an arbitrary number of modes. 
Weaknesses：
This paper will be better if developing a more rigorous mathematical analysis of the approach.

QUESTIONS:
Could you elaborate on the specific implementation details of the geometrical mapping?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03