{"id":"c024cafb-44f5-4442-865e-4545c5a61d7b","arxiv_id":"2504.16262","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"VPFB learns an energy field whose gradient flow matches a Gaussian-to-data homotopy, reaching CIFAR-10 FID 6.72 without MCMC training.","lead":"VPFB trains an energy-based image generator without Markov chain Monte Carlo sampling during training, by matching a learned potential flow to a Gaussian-to-data density path through a variational loss. The method obtains competitive image quality and outlier detection on standard benchmarks, but its central proof contains a circular assumption that needs scrutiny.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Prop. 3 makes the target homotopy equality a premise at C.3 eq. (61), so loss (22) is a local matching condition, not a proven KL minimizer; Section 3.5 imposes the Boltzmann equilibrium by fiat.","rationale":"The reader's weakest-assumption identification points to the same passage in Appendix C.3, and I agree that the hidden premise is load-bearing. I mark partial rather than full agreement because the proof includes an induction argument: if rho_Phi equals bar_rho at time t and the PDE is solved, the one-step KL is minimized, so equality can be propagated from the Gaussian prior. That makes the step a local consistency condition rather than an outright circular definition. The unresolved issue is the absence of any contractivity result: when the trained density is not exactly on the target homotopy, the loss has no proven tendency to reduce the mismatch. This matters because neural network parameterization, stochastic optimization, and the extra terms in the implemented loss (25) guarantee mismatch. The stationarity concern in Section 3.5 is also real: setting p_infinity = bar_rho(t >= t_max) by construction makes the Boltzmann energy approximate the data by design, not by convergence. Several self-acknowledged limitations in the paper support a conditional rather than reject verdict: Appendix B.2 admits that structured Gaussian paths resemble short-run MCMC and can fail in low-density regions, leading to mode collapse under deterministic ODE sampling; Section 4.1 restricts the SDE/ODE consistency to unimodal Gaussian priors; and Section 4.6 shows that long-run ODE sampling degrades, requiring SGLD at inference. These statements are consistent with the theoretical soft spot I identify. The paper has useful empirical results and a plausible variational construction, but the central theorem connecting the loss to KL-minimizing homotopy matching needs repair. I would keep the reader's CONDITIONAL verdict: the theoretical claims should be fixed or weakened, and the empirical claims need code and error bars before acceptance.","tokens_in":28742,"tokens_out":9956,"duration_ms":106112,"concrete_test":"Independently re-derive the first variation in Appendix C.3 without substituting rho_Phi = bar_rho at the line preceding eq. (61), keeping all terms involving rho_Phi and its derivatives. If the resulting optimality condition is not the density-weighted Poisson equation (14), or contains rho_Phi on the left-hand side in place of bar_rho, then Proposition 3 is false and loss (22) is not equivalent to minimizing the KL divergence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that minimizing the VPFB loss (22) matches rho_Phi to bar_rho, and that Phi_B in (20) is the stationary Boltzmann energy. In the proof of Prop. 3, the first variation of the one-step KL is evaluated for a general rho_Phi, but equation (61) then states that rho_Phi(x,t) is assumed equivalent to bar_rho(x,t) before deriving the density-weighted Poisson equation (14). With the base-case induction this is not a flat circle, but it is a local preservation statement: the constructed velocity is the one that keeps rho_Phi on bar_rho provided it is already there. Nothing in the proof shows that the objective (22) contracts the KL when rho_Phi differs from bar_rho, which is exactly the situation for a finite-capacity network trained by SGD. Proposition 6 only guarantees uniqueness of the PDE solution; it does not tie (22) to KL minimization away from the assumption. Section 3.5 then substitutes convergence by declaration, setting p_infinity(x) = bar_rho(x, t >= t_max) approximately p_data, and the implemented loss (25) adds a cosine alignment term, Poincare regularization, and a time-derivative penalty absent from (22). Consequently, the theoretical bridge from the KL objective to the implemented loss, and from the learned potential to the Boltzmann energy (20), is not established. The empirical numbers are encouraging but, without code or repeated-run error bars, they do not close this gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Variational Potential Flow Bayes (VPFB), a framework for training energy-based generative models without contrastive divergence or auxiliary networks. The method parameterizes a potential flow Phi(x,t) and aims to match the flow-driven density homotopy to a data-recovery marginal homotopy built from Gaussian perturbation kernels. The authors derive a density-weighted Poisson equation (Eq. 14) as a stationarity condition, reformulate it as a Deep Ritz / covariance variational loss (Eq. 22), and claim that at stationarity the learned potential equals a Boltzmann energy (Eq. 20) usable for generation, interpolation, compositional generation, and OOD detection. Experiments on 2D synthetic densities, CIFAR-10, and CelebA report competitive FID and AUROC scores, with additional ablations and efficiency comparisons in the appendices.","tokens_in":29177,"tokens_out":7701,"duration_ms":74889,"significance":"If the theoretical claims were fully established, VPFB would be a significant contribution: it offers a sampling-free training objective for explicit EBMs, an interpretable energy landscape, OOD detection as a by-product, and a clear connection between potential flow and diffusion/flow-matching theory. The use of the Deep Ritz method for a density-weighted Poisson equation in generative modeling is novel and interesting. The paper is also unusually honest about its limitations, including long-run ODE divergence, the need for SGLD at inference, and the trade-off between Boltzmann stationarity and sample quality. However, the central theoretical bridge is currently not proven: Proposition 3's proof assumes the homotopy equality it is meant to establish, stationarity is enforced by construction in Section 3.5, and the implemented loss (25) differs from the analyzed loss (22). These gaps mean the paper's headline claims outrun the supporting theory, and the empirical results lack repeated-run statistics. The idea is promising and the empirical core may be salvageable, but substantial revision is required.","major_comments":[{"comment":"Proposition 3's equivalence claim between solving the density-weighted Poisson equation (14) and minimizing the KL divergence D_KL[rho_Phi || rho_bar] is not established. In the proof of Proposition 3, the first variation of the one-step KL is computed for a general rho_Phi, but equation (61) then substitutes rho_Phi(x,t) ≡ rho_bar(x,t) before deriving the Poisson equation. Consequently, equation (14) is a local condition that preserves homotopy matching provided the flow density is already on the target marginal; the proof does not show that minimizing the loss in (22) contracts the KL when rho_Phi differs from rho_bar, which is precisely the situation during training of a finite-capacity network. This is load-bearing because the paper uses Proposition 3 to claim that the variational loss minimizes KL divergence between homotopies (see Section 3.4 and the remark following Proposition 6). The authors should either prove a contraction or bound for the KL under the proposed loss, or explicitly restate Proposition 3 as a preservation property and temper the framing accordingly.","section":"Section 3.2 / Appendix C.3, Eq. (61)"},{"comment":"The stationary Boltzmann equilibrium is imposed by fiat: the text sets p_infinity(x) = rho_bar(x, t >= t_max) ≈ p_data(x) and then equates this to the Boltzmann distribution. As written, 'the stationary Boltzmann distribution approximates the true data likelihood by design', so the learned energy approximates the data by construction rather than by convergence of the learning procedure. Proposition 5's conclusion is therefore conditional on an assumption that is not derived from the flow-matching dynamics and is not empirically verified. This is load-bearing for the claim that VPFB learns an explicit EBM whose energy is a valid OOD score. The authors should provide a quantitative assessment of the discrepancy between rho_bar(x, t_max) and p_data (for example, an FID or likelihood evaluation of the marginal at the cutoff), or clearly label the stationarity enforcement as an additional modeling assumption separate from the theoretical claims.","section":"Section 3.5, Eq. (20)"},{"comment":"The implemented loss (25) differs from the theoretically analyzed loss (22) by the cosine alignment term, the Poincare regularization η||Phi||^2, the time-derivative penalty, and the weighting w(t). No theorem connects this full implemented loss to the weak formulation (21) or to the KL minimization claim of Proposition 3. The cosine term, in particular, changes the scale-invariance properties of the objective, and the Poincare penalty changes the minimizer of the variational problem. The ablation study in Table 3 shows these terms are empirically important, but the theory section only covers loss (22). As a result, the claim that the actual training procedure optimizes the proposed KL/Deep Ritz objective is unsupported. The authors should either extend the theoretical analysis to the full loss (25) or explicitly present (25) as an empirically motivated modification whose relationship to (22) is to be established.","section":"Section 3.4-3.5, Eqs. (22) and (25)"},{"comment":"The paper reports single FID and AUROC values without standard deviations or multiple seeds. FID estimates are known to be noisy, and several reported margins over competitors are small (for example, VPFB's FID 6.72 versus Flow Matching's 6.35, or AUROC 0.78 versus CDRL's 0.75 on CIFAR-10). The checkpoint selection rule (lowest FID on a validation subset, per Appendix D.5) further biases the reported numbers. Since the empirical claims of competitiveness are a major part of the paper's contribution, the authors should provide means and standard errors over at least three independent training runs, or clearly state that the results are single-run and interpret them as preliminary without strong comparative statements.","section":"Tables 1 and 2"}],"minor_comments":[{"comment":"The gradient-norm term in Eq. (25) writes ∇_{(x,t)}Phi, which appears to be a typo for ∇_x Phi; in addition, the time-derivative penalty mentioned in the text is not explicitly displayed in the equation.","section":"Eq. (25)"},{"comment":"The notation for the Sobolev space is inconsistent: Proposition 3 states Phi in H^1_0(Ω,p) while the text earlier defines H^n_0; please harmonize the notation throughout.","section":"Section 3.2 / Appendix C.3"},{"comment":"The diffusion coefficient formula g(t) = -√(2σ(t)(σ̇(t)+f(t)σ(t))) carries a minus sign outside the square root; since g is conventionally positive, please verify the sign convention and define the square-root branch explicitly.","section":"Eq. (16)"},{"comment":"The caption should clarify that the plotted ratio is measured while training without the Poincare regularization, whereas the final models include the Poincare penalty; as written, the figure may be misinterpreted as validating the final loss.","section":"Figure 10"},{"comment":"The column header 'Other Likelihood-based Models' includes GAN-based models such as SN-GAN and BigGAN, which are not likelihood-based; please reorganize the table or adjust the caption to avoid confusion.","section":"Table 1"},{"comment":"The training setup states a terminal time of tend = 1, while Appendix D.3 states that best ODE sampling results use tend = 1.575; please clarify the distinction between training and sampling time horizons.","section":"Appendix D.2 and D.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a first-version arXiv preprint with substantial theoretical gaps in the core equivalence claims. The most serious issue, the assumption in Eq. (61) of Appendix C.3, is clearly visible in the proof and should be caught in review. I see promise in the empirical framework and the Deep Ritz treatment of the Poisson equation, but the authors need to either prove the KL-contraction claim or significantly reduce the theoretical claims to match what is actually shown. The lack of repeated-run statistics should also be addressed. The paper may be suitable for a top ML venue after these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"VPFB is a plausible new way to train EBMs without implicit MCMC, and the empirical story is strong enough that I'd send it to serious referee. But the main theoretical claim does not hold as stated, and the gap matters for how the paper should be read.\n\nWhat is genuinely new: applying the Deep Ritz variational principle to a density-weighted Poisson equation from the feedback particle filter and repurposing that as a generative potential flow. I don't see that combination in the prior work they cite. The paper also deserves credit for honest empirical self-assessment. The ablations isolate the covariance loss, cosine alignment, and Poincaré regularization, and the long-run sampling section openly acknowledges that deterministic ODE sampling diverges in high dimensions and that SGLD is needed for stationary behavior. That transparency is real credit.\n\nThe soft spot is load-bearing. The proof of Prop. 3 in Appendix C.3 starts with a general first variation for the one-step KL, then equation (61) assumes ρ_Φ ≡ ρ_bar before deriving the density-weighted Poisson equation. As written, that makes the result a local preservation condition: the constructed velocity keeps the density on the target homotopy only if it is already there. Nothing in the proof shows that the variational loss (22) contracts the KL when the densities differ, which is precisely the finite-capacity SGD setting. So the statement that solving (14) minimizes the KL between flow-driven and marginal homotopies is not established. Section 3.5 then enforces stationarity by declaring p_∞(x) = ρ_bar(x, t ≥ t_max) ≈ p_data, which makes the Boltzmann link approximate by construction rather than by convergence.\n\nThe implementation widens the gap further. The loss actually trained, Eq. (25), adds a cosine alignment term, a Poincaré penalty, and a time-derivative penalty that are not in Prop. 6's functional (22). Each addition may have a heuristic justification, but the trained objective is not the proved objective. On the empirical side, FID and AUROC are reported without error bars and code is not released, so I cannot tell how repeatable the gains are.\n\nThe citation pattern is fine. The references to flow matching, Deep Ritz, and the particle filter literature are accurate, including the prior density-weighted Poisson solver work. No citation red flags.\n\nWho gets value from this: people working on MCMC-free EBM training, energy-based samplers, or the interface between flow matching and explicit densities. I would not desk reject it. I'd send it to review with a request for major revision: restate Prop. 3 as a local optimality result or prove the stronger contraction claim, align the implemented loss with the stated objective, and provide code plus repeated-run statistics.","headline":"A promising MCMC-free EBM training scheme that earns referee time, but the central theoretical claim overreaches: Prop. 3 assumes the homotopy equality it needs to prove, and the implemented loss differs from the proved one.","tokens_in":29670,"tokens_out":3121,"would_cite":false,"duration_ms":31279,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"VPFB trains an energy-based model by matching a potential-flow density path to the data, replacing MCMC negative sampling with a variational Poisson-equation loss.","keywords":["energy-based models","potential flow","density homotopy","Deep Ritz method","variational principle","flow matching","Poisson equation","out-of-distribution detection"],"falsifier":"On a low-dimensional target with a known density, such as a Gaussian mixture in two dimensions, train VPFB and then estimate the true density of the ODE-transported particles at several intermediate times by histogram or kernel binning; if the KL divergence between that measured flow density and the analytic marginal homotopy $\\bar{\\rho}(x,t)$ does not decrease toward zero as the Deep Ritz loss is minimized, the equivalence claimed in Proposition 3 is contradicted. A second, cheaper check: evaluate the residual of the density-weighted Poisson equation using the learned $\\Phi$ and Monte Carlo estimates of the innovation expectation; a nonzero residual at the converged loss minimum would show the variational loss is not solving the PDE it claims to solve.","tokens_in":28569,"feed_emoji":"⚛️","tokens_out":13645,"duration_ms":113414,"temperature":0.7,"pith_summary":"VPFB (Variational Potential Flow Bayes) aims to show that an energy-based generative model can be trained without the contrastive MCMC sampling that makes standard EBMs unstable and expensive. The method learns a scalar potential whose gradient pushes particles from a Gaussian prior to the data, and insists that the evolving density of this flow match a prescribed interpolating path between prior and data. The paper recasts that matching as a density-weighted Poisson equation and then as a Deep Ritz variational loss, a covariance between the potential and an innovation term plus an expected gradient-norm penalty, so training becomes plain gradient descent. If the argument holds, the steady-state potential is a genuine Boltzmann energy: the same network that generates samples also ranks inputs for out-of-distribution detection, and no auxiliary generator or MCMC warm-up is needed during training.","feed_headline":"Density-flow matching trains energy-based models without MCMC","feed_subtitle":"The learned potential doubles as a Boltzmann energy: one network generates samples and flags out-of-distribution inputs.","key_machinery":"The argument is carried by three linked objects. First, the conditional density homotopy $\\rho(x|\\bar{x},t) \\propto \\exp(\\alpha(t)\\log q(x) + \\beta(t)\\log p(\\bar{x}|x))$, which the paper shows is a Gaussian perturbation kernel $\\mathcal{N}(\\mu(t)\\bar{x}, \\sigma(t)^2 I)$ interpolating between a Gaussian prior and the data-conditioned posterior; marginalizing it over data defines the target path $\\bar{\\rho}(x,t)$. Second, the density-weighted Poisson equation $\\nabla\\cdot(\\bar{\\rho}\\nabla\\Phi) = \\frac12\\,\\mathbb{E}_{p_{\\text{data}}}[\\rho(\\gamma-\\bar{\\gamma})]$, which encodes equality of the flow-driven density and the target homotopy, with innovation term $\\gamma = (\\dot{\\alpha}/\\omega^2)\\|x\\|^2 + (\\dot{\\beta}/\\nu^2)\\|x-\\bar{x}\\|^2$ centered at its conditional mean $\\bar{\\gamma}$. Third, the Deep Ritz loss $L(\\Phi,t) = \\operatorname{Cov}_{\\rho p_{\\text{data}}}[\\Phi,\\gamma] + \\mathbb{E}_{\\bar{\\rho}}[\\|\\nabla\\Phi\\|^2]$, whose first-order optimality condition reproduces the weak form of the Poisson equation; a Poincar\\'e-type spectral gap is invoked to guarantee a unique minimizer. The Boltzmann energy identity $\\Phi_B = (4\\Phi_\\infty + f_\\infty\\|x\\|^2)/g_\\infty^2$ then connects the steady-state potential to the EBM likelihood, and the implemented training loss adds a cosine alignment term between $\\nabla\\Phi$ and the flow-matching conditional vector field plus a quasi-static penalty on $\\partial\\Phi/\\partial t$.","core_discovery":"VPFB's central claim is that training an energy-based generative model reduces to a variational-principle problem: rather than sampling negative examples with MCMC, one learns a potential $\\Phi(x,t)$ whose gradient field transports a Gaussian prior along a density path that matches a prescribed homotopy to the data. The paper proves (Propositions 3 and 6) that solving the density-weighted Poisson equation $\\nabla\\cdot(\\bar{\\rho}\\nabla\\Phi) = \\frac12\\,\\mathbb{E}[\\rho(\\gamma-\\bar{\\gamma})]$ is equivalent to minimizing the KL divergence between the flow-driven density and the target marginal homotopy, and that this is in turn equivalent to minimizing a Deep Ritz loss consisting of a covariance between the potential and the innovation $\\gamma$ plus the expected squared gradient norm. At stationarity (Proposition 5), the same potential parameterizes a Boltzmann energy $\\Phi_B = (4\\Phi_\\infty + f_\\infty\\|x\\|^2)/g_\\infty^2$, so the trained network is simultaneously a sampler and an energy surface for OOD detection. Training freezes the flow near the terminal time and treats the truncated marginal as the data distribution, making the Boltzmann form hold by construction rather than by demonstrated convergence.","pith_inferences":["If Proposition 3's equivalence is as general as stated, the same variational principle should transfer to other perturbation kernels, such as heat-equation blurring or general linear corruptions, by recomputing the innovation term; the paper points to Neumann-boundary blurring kernels as future work, but the variational structure suggests the swap is mechanical.","The Boltzmann interpretation stands or falls with how close the truncated marginal $\\bar{\\rho}(x,t_{\\max})$ is to the data; the paper's AUROC scores vary noticeably across OOD datasets, suggesting the energy's discriminative strength is dataset-dependent in a way the proof does not quantify.","The added cosine-alignment and quasi-static terms in the implemented loss go beyond the variational derivation, so the learned $\\Phi$ is a Boltzmann energy only up to how strongly these extra terms perturb the minimizer; an ablation at fixed FID that isolates their effect on the energy's equilibrium behavior would clarify their role.","Because the covariance loss ties the energy magnitude to the residual of a Gaussian fit, the framework may be sensitive to the Gaussianity of the conditional path; extending the SDE formulation to non-Gaussian or learned priors, which the paper flags as a limitation, would test whether the energy-based interpretation survives beyond unimodal Gaussian transport."],"forward_implications":["EBM training reduces to a gradient-descent objective, a covariance loss plus gradient-norm penalty, with no MCMC negative sampling, so the main cost and instability of contrastive divergence disappears.","The trained potential is an explicit Boltzmann energy: the same network generates samples by ODE or SGLD and assigns energies that separate in-distribution from out-of-distribution inputs, as the paper's AUROC experiments demonstrate.","Because the training path is the optimal-transport flow-matching schedule, particle trajectories are nearly straight, so deterministic ODE sampling reaches competitive FID (6.72 on CIFAR-10 for the time-varying model) at a fraction of the training cost of the MCMC-based EBM baselines reported in the paper.","The MCMC-free promise applies to training: at inference, long-run deterministic ODE sampling drifts into visual artifacts in high dimensions, and the paper finds that stochastic SGLD sampling from the learned Boltzmann energy is what reaches the steady-state equilibrium.","Compositional generation becomes energy averaging: conditioning on a combination of attributes is approximated by averaging conditional potentials, which the paper demonstrates on CelebA attribute pairs."],"supporting_citations":[{"why":"Introduces log-homotopy particle flow, the transport-of-prior-particles idea that VPFB turns into a generative potential flow.","marker":"Daum & Huang (2007)"},{"why":"Supplies the Deep Ritz variational method that converts the density-weighted Poisson equation into a trainable loss.","marker":"E & Yu (2018)"},{"why":"Establishes the SDE and probability-flow-ODE equivalence whose marginal homotopy and stationary solution Propositions 4 and 5 build on.","marker":"Song et al. (2021)"},{"why":"Provides the flow-matching conditional vector field and the optimal-transport schedule used as the VPFB training path.","marker":"Lipman et al. (2023)"},{"why":"Cited for the existence and uniqueness theorem for the weak Poisson formulation that Proposition 6 relies on.","marker":"Laugesen et al. (2015)"},{"why":"Gives the drift and diffusion coefficient formulas linking the Gaussian perturbation kernel to the diffusion SDE.","marker":"Karras et al. (2022)"},{"why":"Originates the density-weighted Poisson equation in the feedback particle filter, the equation VPFB solves variationally.","marker":"Yang et al. (2013)"},{"why":"Supplies the SDE drift and diffusion parameterization used for the OT-FM training schedule.","marker":"Kingma & Gao (2023)"},{"why":"Defines contrastive divergence, the MCMC-based training paradigm that VPFB claims to replace.","marker":"Hinton (2002)"}],"fun_headline_variants":["No MCMC: VPFB trains EBMs via potential flow","Potential flow replaces MCMC in energy-based models","VPFB: One network samples and flags OOD, no MCMC","Variational principle trains EBMs, skips MCMC sampling","Energy models learn by density homotopy, no MCMC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the flow-driven density already equals the target marginal homotopy: the proof of Proposition 3 derives the density-weighted Poisson equation from the KL objective only after assuming $\\rho_\\Phi(x,t) \\equiv \\bar{\\rho}(x,t)$, the very equality the training is supposed to produce, and the stationarity claim similarly treats the truncated marginal at $t_{\\max}$ as the data distribution by construction.","fun_headline_variants_meta":{"raw":{"variants":["No MCMC: VPFB trains EBMs via potential flow","Potential flow replaces MCMC in energy-based models","VPFB: One network samples and flags OOD, no MCMC","Variational principle trains EBMs, skips MCMC sampling","Energy models learn by density homotopy, no MCMC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000243,"raw_usage":{"total_tokens":1556,"prompt_tokens":997,"completion_tokens":559,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":473}},"tokens_in":613,"tokens_out":559,"duration_ms":5293,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:08:47.117485+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a low-dimensional target with a known density, such as a Gaussian mixture in two dimensions, train VPFB and then estimate the true density of the ODE-transported particles at several intermediate times by histogram or kernel binning; if the KL divergence between that measured flow density and the analytic marginal homotopy $\\bar{\\rho}(x,t)$ does not decrease toward zero as the Deep Ritz loss is minimized, the equivalence claimed in Proposition 3 is contradicted. A second, cheaper check: evaluate the residual of the density-weighted Poisson equation using the learned $\\Phi$ and Monte Carlo estimates of the innovation expectation; a nonzero residual at the converged loss minimum would show the variational loss is not solving the PDE it claims to solve.","supporting_citations":[],"review_version":1}