Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This work proposes a score-based method for learning neural networks to model wavefunctions minimize quantum potentials. The work adapts an objective function from the VMC model that minimizes local energy under current model samples. The main innovation is the proposal to model only the gradients of the log wave function instead of the wave function itself, which is motivated by an insight into the functional form of the local energy. DMC weighting is introduced to overcome optimization difficulties that arise when the wavefunction is not directly available. The proposed method is evaluated on a boson potential and a challenging fermion potential, and experimental results show a good match to physical observations.
strength_and_weaknesses: STRENGTHS:

1. The work is based on an interesting insight into the functional form of the wavefunction that appears in the local energy loss. Replacing the gradient of an explicit wavefunction with a function that directly estimates the score function is a compelling proposal that is aligned with similar developments in generative modeling.
2. The hybridization of DMC and VMC is an interesting solution to the optimization difficulties that arise when replacing the explicit wave function with the score function. DMC provides a way to move towards states with low potential, while VMC provides a more principled probabilistic proposal for DMC based on the score matching objective.
3. The work explores an exciting area at the intersection of deep learning and computational physics.

WEAKNESSES:

3. The Ferminet network used for experiments already provides an estimate for $\psi_0$, so the motivation for score-based modeling is not as clear. What is the motivation or benefit of learning the gradient of a scalar-valued net, instead of learning a network that outputs the estimated gradient directly as typically done in score-based modeling? The fermion restrictions are a difficult challenge with limited options available and the proposed method might be useful for future architectures, but at the present it seems roundabout to use score-based modeling for a scalar valued net.
4. The experimental results appear somewhat limited, where only a Gaussian-like density and fermion density are investigated. However, I am not familiar with the quantum physics literature and cannot provide perspective on the scope or significance of the results. The fermion results do not match the performance of Ferminet, which might limit the appeal of the proposed method especially since the same architecture is used.
5. While interesting, the formulation of the WSM objective might be somewhat ad-hoc. Is there a more principled way to integrate DMC and VMC beyond an intuitive connection?

clarity,_quality,_novelty_and_reproducibility: Suggestions for making the paper more accessible to those without physics expertise:

1. The relation between the WSM objective and V could be explained intuitively to help reader understanding. From what I gather, the score model initially points in many random directions, and the weightings gradually guide the score function to develop "local minima" (for vector fields, this would mean regions where the score function "points to") where the potential V is minimized by causing the model to gradually see many samples of states with lower potential energy. Is it correct to say that the potential V affects learning only through the weightings? It would be helpful to compare the role of V in the proposed work and existing works that have an explicit wavefunction.
2. The exact objective of the method could be stated more clearly. The second-to-last paragraph of page 2 starts with "Our objective is to find the ground state $\psi_0$) but the method appears to learn the gradient of the ground state and samples from the ground state. What is the exact primary goal? I suppose that samples from the ground state wavefunction are the practical objective. If the goal is simply to draw samples from the minimum-energy ground states, why not optimize $V(x)$ directly? These questions are likely naive but could help other non-experts.
summary_of_the_review: I enjoyed reading this paper and appreciate the efforts to develop deep learning methods that are useful in computational quantum physics. I have little knowledge of quantum physics so I can only comment on ML aspects of the paper. Replacing the explicit wave function with a score network is technically relevant and mirrors similar developments of EBM and score models in deep generative modeling. The integration of VMC and DMC is interesting but would be stronger with a more principled formulation if possible. I cannot provide any context on the quality or significance of the experimental results. A major experimental weakness is that the work uses a network with explicit wavefunction for the central experiment even though the motivation of the work is to avoid explicitly modeling the wave function. It seems there is currently no suitable architecture for the proposed method. Overall, I recommend the paper, but I am on the fence.
review_title: null

final_score: rating: 5: marginally below the acceptance threshold
review_content: summary_of_the_paper: The paper proposes a method for solving the quantum many body problem using neural networks as variational ansatz. In contrast to more standard approaches which model the wave function directly, authors use a score-based model (hence modeling the gradient of the underlying probability distribution associated with the wavefunction). Authors draw on techniques from score based modeling and diffusion Monte Carlo to formulate a schema for training such models. They demonstrate that the proposed approach works well for the bosonic quantum harmonic trap. Authors also achieve good results for fermionic atomic systems by using gradients of a FermiNet as a model for the score function.
strength_and_weaknesses: Strengths:
* The paper studies an underrepresented approach to modeling quantum states based on gradients of the logarithm of the wave function amplitude.
* Paper explains methodology clearly and accurately

Weaknesses:
* The proposed method seems to be most suited for problems without the “sign problem”. In this setting diffusion monte carlo and similar methods already provide very good results and then at the very least should be considered as a baseline.
* Some approximations made to set up the optimization loop are not fully justified (or I might have missed the argument, e.g. eqs. 14->15;)

clarity,_quality,_novelty_and_reproducibility: The paper is clearly written and explains relevant concepts well. The article includes relevant references and citations. I was not able to locate all of the details to assess how reproducible the results are (presumably in the appendix which was not included with the main pdf?). Code availability is not stated.
summary_of_the_review: I enjoyed reading the paper and I think the direction taken by the authors is both promising and interesting. Results and ideas are presented clearly and concisely. Authors honestly point out challenges associated with nodal structure in fermionic systems.

While I’m very excited about the field and the research direction considered in this paper, I’m a bit concerned that the proposed method addresses problems that are already “solved” by classical techniques such as diffusion Monte Carlo (DMC). In most cases the resort to parametric variational Monte Carlo (VMC) methods is motivated by the sign problem which results in poor estimates in DMC techniques. This is however an exciting direction and would be of great interest to a more narrow audience in computational physics (and a potentially major milestone if such method could be effectively generalized to fermionic systems). Hence I’m leaning towards the score of (5), but could be convinced otherwise.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The authors propose Diffusion Variational Monte Carlo (DVMC) for simulating many-body quantum systems based on the neural score function. Contrast to a conventional Variational Monte Carlo (VMC), the proposed method requires estimating the score function only, which is a gradient of the logarithm of the wavefunction, rather than evaluating the wavefunction itself. To this end, the authors formulate a new loss function, named weight score matching (WSM, which is analogous to the score matching used for energy-based models (EBMs)), that can learn a ground state of the target wavefunction with the score function only. By combinding such a loss function with the Lagevin dynamics, the proposed DVMC framework samples and learns the targeted ground state simultaneuosly. The authors validate their approach for two quantum systems.
strength_and_weaknesses: Strength:

1. The motivation of the propsed framework, i.e., learning and sampling ground states by using the score function only, is clear. First, MCMC step of VMC learning can be removed. It takes a similar advantage as by-passing expensive MCMC likelihood estimation in the score matching method of EBMs. Second, as the authors mention, one can omit the necessity of computing the score function from the wavefunction.

2. The authors introduce a permutation-equivariant architecture [1] to guarantee the indistinguishability of quantum mechanical particles. To me, it is a novel application of [1] (although not sure because I am not an expert of machine-learning-based quantum simulations).

Weakness:
1. The authors derive the local energy of QMC, and conclude its kinetic term is equal to the implicit score matching (ISM) [2] (up to a constant). However, it is not true considering a coefficient of the norm of the score function $||s_{\theta}(\mathbf{x})||^2$. Note that the kinetic term of the local energy of QMC is given by $\text{tr}(\nabla_{\mathbf{x}}s_{\theta}(\mathbf{x})) + ||s_{\theta}(\mathbf{x})||^2$ while the original ISM is $\text{tr}(\nabla_{\mathbf{x}}s_{\theta}(\mathbf{x})) + 0.5||s_{\theta}(\mathbf{x})||^2$. Thus, it is not entirely clear the connection between the local energy and ISM. 

2. It seems that the major advantage of the proposed DVMC is its learning and sampling efficiency. However, there is no benchmark on the computational efficiency of DVMC.

3. The authors introduce four assumptions (in page 5) to eliminating the wavefunction-dependency and distributional shift of the porposed framework. However, they do not dicuss whether the introduced assumptions are reasonable (acceptable) or not.

4. For fermion systems, the proposed method requires computing the wavefunction. Therefore, the main claim of this paper is partially applicable (only for boson systems). 

5. Experimental results are weak. The harmonic trap example is very simple and can be solved analytically. Atomic systems (B/C/N/O) are more intersting, but they are fermion systems, which doe not support the main claim of this paper as I mentioned in 5.

6. The authors calim "numerous examples in machine learning have shown that implicitly modeling the data distribution provides better representation" and "properly chosen latent embedding can better describe the struvture of the data space" with refering VAE and GAN literatures in page 1 of the paper. Because the used algorithm (score matching) is not based on the latent representation and the used data (quantum state) seems to be not an embedded manifold, such claims are not entirely clear to me.

Some minor issues:
1. In (3), it seems that $\mathbf{x} \sim \psi_{\theta}^{2}(\mathbf{x})$ should be $\mathbf{x} \sim \psi_{\theta}^{2}(\mathbf{x}) / \int \psi_{\theta}^{2}(\mathbf{x}) d\mathbf{x}$.
2. In (7), the ISM should be $\mathbb{E} \[ \text{tr}(\nabla_{\mathbf{x}}s_{\theta}(\mathbf{x})) + 0.5||s_{\theta}(\mathbf{x})||^2 \]$.
3. In page 6, the authors mention "The equivariance can be easily achieved by considering the input coordinates as a set". Please elarborate it more detaily or add a reference (might be [1]?)
4. The potential $V(\mathbf{x})$ is not included in (18 - 19).

** *
[1] Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., & Smola, A. J. (2017). Deep sets. Advances in neural information processing systems, 30.

[2] Hyvärinen, A., & Dayan, P. (2005). Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4).

clarity,_quality,_novelty_and_reproducibility: Clarity:

The paper is generally easy-to-follow, but has some mathematical typos as I mentioned in the above section. The main topic of this paper, i.e., the connection between the local energy and ISM, is not entirely clear for its current form.

Quality:

The motivation of this paper is clear. However, experiments do not support the motivation (by-passing the wavefunction evaluation and enhancing the computational efficiency)

Novelty:

An introduction of the score matching for QMC is novel for me. However, my evaluation might be not exhaustive because I am not an expert on QMC field.

Reproducibility:

The paper contains an incomplete set of used model architectures and hyper-parameters. Code is not made publicly available.
summary_of_the_review: Although the paper is interesting, I think there is still much room for improvement. I think the paper is not ready for publication in its current form.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: Consider the problem of finding the ground-state of the Hamiltonian $H = -\frac{1}{2} \nabla^2 + V$, i.e. its lowest eigenfunction $\psi$. This corresponds to minimising the loss function $\mathcal{L}(\theta) = \mathbb{E}_{|\psi|^2}\left[ E(x,\psi) \right]$ where $E(x, \psi) = \frac{H \psi(x)}{\psi(x)}$ and the wave-function $\psi$ is parametrised with a neural-network with weight $\theta \in \Theta$. We have that

$$ \nabla \mathcal{L}(\theta) = \mathbb{E}_{|\psi|^2}\left[  \left(E(x,\psi)  -  \mathcal{L}(\theta) \right)  \nabla \log |\psi(x)|  \right].$$

Instead of parametrising the wave function, the articles proposes to parametrize (half-of) the score $s(x) = \nabla \log |\psi(x)|$ of the probability density proportional to $|\psi^2|$. In order to learn the score of the ground-state, the proposed approach maintains at iteration $t \geq 0$ an approximation of the score function $s^{(t)} = \nabla \log |\psi^{(t)}|$ as well as a population of $N$ particles $[x^t_i]_{i=1}^N$ approximately distributed according to $|\psi|^2$.

1. at iteration $t \geq 0$, for a current approximation of the score $s^{(t)}$ and population $[x^t_i]_{i=1}^N$, use (approximate) Langevin dynamics with $s^{(t)}$ and finally compute the usual Diffusion-Monte-Carlo (DMC) weights. This procedure gives a slightly better approximation of the ground state expressed as a weighted population of particles (i.e. it is the working principle of DMC)
2. Instead of resampling with these weights as a standard DMC method would do, use (weighted) score matching to get an approximation go this weighted population of particles. Since the weighted population of particles was a better approximation of the ground state, this results in a better estimate of the score of the ground state.
3. iterate


strength_and_weaknesses: **Disclaimer:** I am not an expert in ML methods for computing the ground state of quantum systems. I am **not** on top of the current literature.

**Strength:** the paper is accessible and reads reasonably well. I believe that the fundamental motivation for the proposed methodology is the claim that modelling the score of the ground state distribution is somehow easier than modelling the distribution itself (or equivalently, parametrising the score is easier that parametrising the wave function with a neural net). I am finding it interesting that the authors are trying to study this claim. I do not know the literature well enough to comment on the novelty of the approach.

**Weakness:**
1. parametrising the score (instead of the wave function) means that it is not possible to use an exact Metropolis-Hastings (MH) accept/reject. The authors choose to completely ignore the problematic quantities within the MH and I am finding this step a bit hard to swallow. More comments and simulations would be welcome.
2. All the quantities in Equation (4) can be expressed in term of the score: the function $E$ is a function of the score, and the gradient term is exactly the score. Why not discretize this quantity to get an approximation of $\nabla \mathcal{L}(\theta)$ and update the score? Doing this, we do not even need to compute the DMC weights, no?
3. In order to substantiate the claim that modelling the score is better than directly modelling the wave function, the authors should run the method that consists in evolving a population of $N$ particles with the Langevin-dynamics with **exact** accept-reject, with the same computation of the DMC weights, and update of wave function through Equation (4).
4. It is a bit disappointing that the authors are still relying (as they readily admit it) on the parametrisation of the wave-function (i.e. FermiNet parametrisation), which seems to defeat the whole purpose of the whole study. 

Again, I would like to re-iterate that I am not an expert in this domain (and may have mis-understood some aspects of the proposed method) and I am looking forward to hearing the authors comments.
clarity,_quality,_novelty_and_reproducibility: **Clarity:**
as a non-expert, I have been able to easily understand the text. The text is clear.


**Quality:** I am not finding the text extremely convincing when it comes to the main claim that modelling the score is ``better" than modelling the wave-function itself. This may be true, in some settings, but I have not finding the text, as well as the numerical simulations, extremely convincing. Again, I may have mis-understood some of the aspects of the proposed methodology. I am looking forward to the authors' response.


**Novelty:** I do not know the modern literature on this problem well-enough to comment.


**Reproducibility:** there are enough details provided in the main text to reproduce the experiments.


summary_of_the_review: In many settings, modelling the score of a distribution has advantages when compared to modelling the distribution itself (eg. generative modelling). The text proposes to investigate this claim in the context of finding the ground state of the Schrodinger equation.

After reading the text, it is not entirely clear to me whether modelling the score has advantages over modelling the wave-function itself. Still,  the text seems to demonstrate that this approach is viable and deserves more numerical investigations.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The paper is located in the research field of neural wavefunctions, i.e. aiming to compute the ground state energy of quantum many-body systems. While recently works which explicitely represent the wavefunction have shown great success, the author propose to follow an implicit approach via a score-based principle. Given the recent success of this area in other domains, this indeed is an exciting direction to investigate. Overall, all reviewers liked the paper and the idea/approach. Two limitations have been identified and discussed in detail among the reviewers:

1) While the authors main aim is to use an "implicit" approach, the final solution they use is basing on the FermiNet architecture (i.e. again using an explicit approach). This is, to some degree, defeating the entire purpose of the model. The authors address this point partially in their feedback -- and also argue that other architectures have now shown better results. Still, it would be interesting to understand why is this the case. And whether this means that explicit models are actually "better" for the considered scenario.

2) The considers scenarios are a bit simplistic -- and the actual benefit, e.g., on the frequently used molecules is not very clear. (E.g. also the runtime/complexity of the method is not better.)

Overall, even after consultation with all reviewers, the paper is borderline.
justification_for_why_not_higher_score: The paper is clearly borderline and an accept as poster can also be considered.
Decision: Reject