Title: High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction
Venue: NeurIPS 2025
Accepted: yes
Keywords: AI for sciences

Machine learning for science

Hamiltonian prediction

Density functional theory

Quantum physics

Physics

Computational chemsitry

Flow matching

Generative model
Abstract: Density functional theory (DFT) is a fundamental method for simulating quantum chemical properties, but it remains expensive due to the iterative self-consistent field (SCF) process required to solve the Kohn–Sham equations. 
Recently, deep learning methods are gaining attention as a way to bypass this step by directly predicting the Hamiltonian.
However, they rely on deterministic regression and do not consider the highly structured nature of Hamiltonians.
In this work, we propose QHFlow, a high-order equivariant flow matching framework that generates Hamiltonian matrices conditioned on molecular geometry. Flow matching models continuous-time trajectories between simple priors and complex targets, learning the structured distributions over Hamiltonians instead of direct regression. To further incorporate symmetry, we use a neural architecture that predicts SE(3)-equivariant vector fields, improving accuracy and generalization across diverse geometries. To further enhance physical fidelity, we additionally introduce a fine-tuning scheme to align predicted orbital energies with the target. QHFlow achieves state-of-the-art performance, reducing Hamiltonian error by 71% on MD17 and 53% on QH9. Moreover, we further show that QHFlow accelerates the DFT process without trading off the solution quality when initializing SCF iterations with the predicted Hamiltonian, significantly reducing the number of iterations and runtime.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nDeep learning approaches for prediction of the Hamiltonian matrix offer a way to improve the efficiency of density functional theory (DFT) calculations by forgoing or reducing the number of self-consistent field (SCF) steps needed. The authors propose a SE(3)-equivariant flow matching model for the prediction of this Hamiltonian. This is achieved through the introduction of a symmetry-aware prior distribution and orbital energy finetuning procedure. On several datasets, this method demonstrates improved performance in Hamiltonian error metrics, and reduction in SCF steps compared to prior work that predicts the Hamiltonian through regression.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n* *Clarity*: the paper is generally well written and easy to understand.\n* *Scope*: the problem is well defined, and the work includes two distinct benchmark tasks.\n* *Originality*: while SE(3) equivariant flow matching models have been proposed in the past, this work is the first to apply them to the task of Hamiltonian prediction, which has generally been done through direct regression. This is non-trivial due to the higher-order nature of the Hamiltonian matrix.\n* *Significance*: the proposed method offers an improved accuracy of both Hamiltonian prediction and SCF step reduction compared to prior work.\n\n### Weaknesses\n * The authors make several modifications to the QHNet architecture in their method, including the additional input of the overlap matrix $\\mathbf{S}$, attention-based message-passing layers, and a channel mixing module. Furthermore, the introduced energy alignment fine-tuning could also be applied to the regression-based model. Given the focus of the work is on the proposed flow-matching method, it is unclear if the performance gains are due to the generative task or improvements in model architecture.\n\nQUESTIONS:\n* How much of the performance improvement over QHNet is due to architectural changes (see *Weaknesses*) vs. the introduction of flow matching? It would be helpful to expand the ablation studies in section 5.4 to include either a QHNet baseline with the architectural improvements and energy alignment fine-tuning, or a QHFlow implementation without the architectural improvements, so we can better understand the source of the improved performance.\n* Fig. 3: Given QHFlow is evaluated with \"3-step ODE-based sampling\" wouldn't we expect the inference time ratio for QHFLow to be roughly 3 times that of QHNet? While the bar colors of \"Inf T Ratio\" can be inferred by order, it would be helpful to modify this figure so that this is more obvious.\n* What datapoints are the \"71%\" M17 and \"53%\" QH9 Hamiltonian derived from? The authors should be more precise with these statements in both the abstract and results section. Measuring the mean % improvement over the most recent state-of-the-art method would make the most sense to me.\n\nLIMITATIONS:\nThe authors do not include any limitations of their work. One notable limitation in this line of research is that Hamiltonian prediction error is not always well correlated with SCF step reduction. I.e. a model that has a very low $\\mathbf{H}$ MAE might not necessarily improve DFT runtimes, which could suggest a different task if SCF step reduction is the main goal. This does not seem to be the case in this work (at least in aggregate) but could be worth discussing (along with other things).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a flow-matching framework for generating Hamiltonians corresponding to ground-state solutions of DFT. Namely, the method builds on the QHNet architecture while adopting a generative approach in place of direct regression. To ensure SE(3) equivariance during flow matching, the method introduces two SE(3)-invariant priors: a Gaussian orthogonal ensemble and a tensor expansion-based prior. Additionally, the model extends their approach with a post-hoc fine-tuning step using a weighted alignment loss to improve the accuracy of orbital energy predictions. Experimental results on the MD17 and QH9 datasets demonstrate that the method outperforms existing regression-based baselines across multiple metrics.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper is well written and clearly structured.\n\n2. The proposed approach provides SOTA results by leveraging flow matching with equivariance and tailored prior to predict Hamiltonian matrices for the ground-state solution of DFT. The incorporation of SE(3)-invariant priors (GOE and TE) are novel and well-motivated by the symmetry properties of Hamiltonians.\n\n3. The experiments are comprehensive and rigorous, comparing QHFlow with other leading regression-based approaches (e.g QHNet, WANet, SPHNet) across various datasets. The various experiments and metrics strengthen the validity of the findings.\n\n**Weaknesses**\n1. While QHFlow demonstrates strong performance on small to medium-sized systems (e.g., MD17, QH9), including out-of-distribution results within QH9, its scalability to larger and more complex molecules such as those in MD22 is untested. Evaluating larger molecules would help establish applicability to larger chemistry workloads and help strengthen the paper even further.\n\n2. QHFlow primarily extends QHNet with the flow matching and symmetry-aware priors. Although, these proposed components lead to significant empirical improvements, the novelty of the approach is limited as it largely builds on existing methodologies and does not significantly advance methodology.\n\nQUESTIONS:\n**Questions**\n1. How does the method perform (only inference) on even larger systems e.g. Alanine tripeptide (42 atoms) and DHA (56 atoms) from the MD22 dataset?\n\n2. Is there any intuition why the TE prior generally performs better than GOE prior?\n\n3. Line 38-39: “However, these approaches rely on pointwise regression and often fall short 39 of capturing the full distributional structure and uncertainty inherent in the Hamiltonian.” Could the authors clarify what is meant by \"distributional structure and uncertainty\" in this context? To my understanding, there is generally a one-to-one mapping between structure and the ground-state solution, and it is unclear what uncertainty is referring to here.\n\n**Suggestions**\n1. Perhaps I missed it, but it would be helpful to explicitly state that Table 3 reports results from QHFlow using the TE prior. I had to match the numbers between tables to infer this connection.\n\n**Typos**\n1. Table 1: Malonaldehyde -> Malondialdehyde\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 2, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce QHFlow, a flow matching method for predicting converged Kohn-Sham density functional theory (KS-DFT) Hamiltonians. The authors introduce two novel SO(3)-invariant base densities in the same vector space of KS-DFT Hamiltonians. This enables the authors to train a flow matching model conditioned on the molecular structure mapping from one of the base densities to the Dirac distribution of the converged Hamiltonian. In their experiments, the authors find that their QHFlow reduces MAE errors significantly compared to direct Hamiltonian predictors.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper is well written, easy to follow, and provides the necessary background for understanding their work.\n* The authors propose an interesting application for flow matching-based regression.\n\nWeaknesses:\n* While the empirical performance motivates the work very well, the qualitative reasoning provided is weak. For instance, in line 39, it reads \"[...] approaches [...] often fall short of capturing the full distributional structure and uncertainty inherent in the Hamiltonian.\" With the exception of degenerate states, there's neither a distribution of Hamiltonian nor an uncertainty for a given structure. As you correctly note yourself in l.169, where you describe it as a Dirac distribution.\n* The technical novelty of this work is constrained to the two new base densities.\n* Hyperparameters vary quite a bit between datasets, and suggestions such as the WA Loss provide inconsistent results.\n\nQUESTIONS:\n* Figure 3 confused me as the caption and the appendix read that the SCF T Ratio is the ratio of converging a DFT calculation starting from QHFlow vs minao. Though, then a factor of 46 would mean that the QHFlow initialization takes 46 times as long to converge as minao? However, the text reads it's 13-15% faster? An explanation on how the figure should be read would be greatly appreciated.\n* How sensitive is your approach to the choice of hyperparameters? Is there a good way to estimate these a priori?\n\nLIMITATIONS:\nNo, the authors could discuss limitations such as scalability. It appears that the method does not scale to large systems, as the largest ones included here are from QM9.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nDFT is an approximation to solve the Schroedingers Equation. It has received nobel prizes. DFT is computationally demanding and takes hours of compute for a single molecule. In the past, starting with Snyder et al PRL 2012 ML approaches have been used to estimate DFT densities. Other work followed. Also a direct prediction of the Hamiltonian has been proposed (Schütt et al 2019). In the meantime a number of approaches have been conceived. Also some that attempt to directly improve and accelerate the DFT loop. \n\nThe present ms proposes to use higher-order flow matching for DFT level Hamiltonian prediction in an equivariant fashion and presents impressive improvements over SOTA. Also SCF iterations in the DFT loop can be accelerated using the proposed approach.\n\nSTRENGTHS AND WEAKNESSES:\nstrength\n\nThe paper is clearly written, with well-structured sections and precise language that make it easy to follow the authors' arguments and methodology. The clarity of presentation contributes significantly to the overall readability and accessibility of the work.\n\nThe experimental section is impressive. The authors conduct a thorough and well-designed set of experiments that convincingly support their claims. The results are clearly presented, and the comparison with relevant baselines demonstrates the effectiveness of the proposed approach.\n\nThe paper is also well-situated within the existing body of literature. The authors provide a comprehensive and thoughtful review of related work, and they effectively position their contribution in relation to previous studies. This contextual grounding enhances the reader’s understanding of the paper’s novelty and relevance.\n\n\nweaknesses \n- it remains unclear to reviewer from the ms what the differences in the tables mean chemically, i.e. are the improvements insightful or chemically meaningless?\n(To give context to this question: many newer ML approaches in MLFFs provide improvements that are chemically meaningless and yield unstable MDs)\n- I haven't found limits discussed, please discuss them carefully\n\nQUESTIONS:\nOne area that could benefit from further clarification is the chemical meaning of the observed progress. While the results are promising, it would strengthen the paper to elaborate more explicitly on what the improvements represent in terms of underlying chemical principles or phenomena. Providing a clearer interpretation from a chemical standpoint would help bridge the gap between the computational findings and their practical or theoretical implications in chemistry.\n\nAdditionally, the paper would be improved by a more detailed discussion of the limitations of the proposed method. While the results are strong, every approach has its constraints, and a candid evaluation of where the method may fall short—whether in terms of scalability, generalizability, computational cost, or domain-specific applicability—would offer a more balanced perspective. Such a discussion would not only increase the transparency of the work but also guide future research directions.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}