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REVIEW 4 major objections 7 minor 56 references

Optimization Landscapes Learned: Proxy Networks Boost Convergence in Physics-based Inverse Problems

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Optimizing on a neural-network-smoothed copy of a chaotic loss landscape gets BFGS to the true inverse parameters more often than optimizing on the raw landscape.

desk verdict A plausible surrogate-optimization idea undercut by suspicious hyperparameter selection and an ambiguous loss-penalty; deserves referee time but not yet a citation. read the letter →

arxiv 2501.16573 v1 pith:BDWV4BOE submitted 2025-01-27 cs.LG math.OC

Girnar Goyal , Philipp Holl , Sweta Agrawal , Nils Thuerey This is my paper · ORCID
classification cs.LGmath.OC
keywords inverseproblemsconfigurationlosslandscapeproxyneuralnetworkregularizationBFGSpartialdifferentialequationschaoticsystemsbilliards
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the failure of gradient-based optimizers on physics inverse problems stems from the chaotic, multi-minima shape of the configuration loss landscape rather than from the inverse problem itself. It trains a proxy neural network to predict this loss from a target trajectory and a candidate parameter set, then regularizes the network during training to flatten the landscape while keeping the global minimum in place. A two-step scheme—first BFGS on the smoothed proxy loss, then BFGS on the true loss—converges to the true parameters more often than BFGS applied directly to the true loss on Burgers, Kuramoto-Sivashinsky, and billiards problems. The reader should care because this turns a learned surrogate into a tool for optimization, not just prediction.

What carries the argument

The configuration loss $L(Y^*, X_s) = \|\mathcal{P}(Y_0, X_s) - Y^*\|_2^2$ measures how far a trajectory evolved under candidate parameters $X_s$ is from the observed target trajectory $Y^*$. The proxy network $f_\theta(Y^*, X_s)$ is trained to predict $L$ using Adam on a regularized objective $L^R_N = \mu \|f_\theta(Y^*, X_s + \sigma N) - L(Y^*, Y_s)\|^2$, where $\sigma N$ adds Gaussian input noise and $\mu > 1$ penalizes predictions above the true loss. These two pressures smooth high-frequency features of the learned landscape while favoring low-lying regions near minima. The optimization phase runs BFGS first on the proxy-predicted landscape, then on the true configuration loss starting from the proxy's result.

What would settle it

Choose a single inverse problem and vary the noise scale $\sigma$ until the proxy-predicted global minimum moves more than one BFGS tolerance away from the true parameters; the two-step method will then return a wrong answer even with unlimited secondary iterations. A direct check is to plot the proxy landscape's argmin against $\sigma$ and look for a systematic drift.

Watch

Extended reading notes

Core claim

Proxy neural networks can replicate the configuration loss $L(Y^*, X_s) = \|\mathcal{P}(Y_0, X_s) - Y^*\|_2^2$ across spatio-temporal trajectories, and with noise and loss-penalty regularization they produce smoother versions of the landscape whose global minimum still tracks the true parameters. Optimizing this smoothed proxy with BFGS lands near the true optimum, and a second BFGS step on the ground-truth loss refines the result. In the 2D billiards setup, convergence accuracy almost doubles compared with BFGS on the ground-truth loss.

Load-bearing premise

The regularized proxy must keep the global minimum at the true parameters while smoothing the landscape; if the smoothing shifts or erases that basin, the primary BFGS step lands in the wrong region and the secondary step cannot recover.

Editorial extensions

If this is right

  • The two-step proxy optimization improves convergence accuracy over BFGS and gradient descent on the ground-truth loss across all three tested systems (Burgers, Kuramoto-Sivashinsky, 2D and 4D billiards).
  • In the 2D billiards setup, convergence to the optimal solution almost doubles when the primary step uses the proxy-predicted loss rather than the ground-truth loss.
  • Resimulation error, the L2 distance between trajectories from predicted and true parameters, is lower for proxy-optimized parameters than for baseline optimizers.
  • The regularization strength controls a trade-off: too little leaves the chaotic landscape intact, too much produces an oversimplified landscape with poor convergence.
  • ProxyNNs trained with Fourier feature inputs generalize well enough to predict loss values for unseen target trajectories sampled from the same initial state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper does not pursue is to apply the same two-step idea to other optimizers (e.g., Adam or Newton methods) by training the proxy to output a smoothed loss for any candidate parameters, effectively adding a tunable smoothing knob to the optimizer.
  • The asymmetric loss penalty ($\mu > 1$ when the proxy over-predicts) biases the learned landscape toward its low regions; a symmetric penalty or a gradient-norm penalty might yield the same smoothing with different trade-offs between smoothness and basin preservation, which the paper does not test.
  • Because the proxy is trained in tandem with the numerical solver, the method inherits the solver's accuracy; coupling the primary step to a differentiable solver would allow end-to-end gradient flow from the coarse-smoothed landscape to the fine true loss.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The paper proposes ProxyNN, a neural network that learns the configuration loss landscape of physics-based inverse problems, with implicit and explicit regularization (input noise and a loss penalty) intended to smooth the landscape. A two-step optimization scheme is then used: BFGS is applied first to the regularized ProxyNN-predicted loss, and the resulting point is refined by a second BFGS run on the ground-truth configuration loss. The method is evaluated on three inverse problems: the inviscid Burgers equation, the Kuramoto-Sivashinsky equation, and 2D and 4D billiards setups. The authors report that ProxyNN-based optimization improves convergence accuracy over BFGS and gradient descent on the ground-truth loss, with the strongest claim being that convergence 'almost doubles' in the 2D billiards setup.

Significance. If the results hold, the paper would offer a conceptually appealing way to leverage neural network regularization to ease optimization of chaotic, multi-modal inverse loss landscapes, and it identifies a concrete two-step optimization protocol that could be useful in PDE-constrained inverse problems. The manuscript formulates a clear generalized configuration-loss framework, evaluates on several nontrivial benchmarks, and transparently acknowledges in Section 6 that the relationship between regularization hyperparameters and landscape complexity is empirical. The central idea is plausible and the experimental evidence is suggestive, but the evaluation protocol as currently written leaves the strength of the claim uncertain and would need to be tightened for the paper's conclusions to be fully supported.

major comments (4)
  1. [§2.3, Eq. (4)] The regularization described in the prose does not match Eq. (4). The text states that the training loss is scaled with µ > 1 for all samples where fθ(Y*, Xs) > L, which is supposed to create a geometric bias favoring low-lying regions of the loss landscape. Eq. (4), however, multiplies the entire squared error by µ, which for a fixed µ is a constant rescaling of the loss and cannot create the described bias toward regions where L is small. If the implementation used a conditional penalty, the equation must be corrected; if it used Eq. (4) as written, the claimed mechanism for loss-penalty regularization is not present in the paper. The actual regularization used in the experiments is therefore underdetermined.
  2. [§3.3, §4, Table 1] The reported convergence improvements are based on the 'top-performing regularized proxy network' for each system, with the corresponding {σ, µ} listed in Table 1. No validation split or selection rule is described, so the hyperparameters appear to have been chosen by looking at the test-set accuracy curves in Figure 5. This makes the headline improvement (e.g., 'almost doubles' in 2D billiards) the result of post-hoc selection on the benchmark set rather than evidence for a general property of smoothed proxy landscapes. A principled selection criterion (e.g., choose σ and µ on a separate validation subset, or by a criterion that does not use the known X*) and then reporting accuracy on held-out problems is needed to support the central claim.
  3. [§3.3, §4, Figures 4-6] All optimization accuracy numbers are point estimates. There are no error bars or multiple seed runs for the ProxyNN training or for the BFGS optimization, even though the training involves stochasticity (Adam, random sampling of Xs) and the results are likely sensitive to initial guesses. Without a measure of variance, it is unclear whether the reported advantage of ProxyNN over BFGS, which is described as 'almost doubles,' is statistically significant. The manuscript should report mean and standard deviation across at least several independent training seeds and initial-guess choices.
  4. [§2.4, §3.3] The initial-guess protocol for BFGS is not specified. For the 256 unique inverse problems in each setup, it is not stated how the starting point is generated for (i) the baseline BFGS on the ground-truth loss, (ii) the primary BFGS step on the ProxyNN loss, and (iii) the secondary step. If the initial guess for the baseline is not the same as that used in the ProxyNN pipeline, the comparison may be unfair. The manuscript must state the initialization distribution and confirm that the same protocol is used for all methods.
minor comments (7)
  1. [§3.1] The text contains 'Burgerséquation' with a stray accent; it should read 'Burgers equation'.
  2. [§3.2] The sentence 'A summarize the network configurations...' is ungrammatical; it should read 'Appendix A summarizes the network configurations...'.
  3. [Eq. (4)] The second argument of L is written as Ys, while Eq. (1) defines L as a function of Xs; please make the notation consistent throughout.
  4. [Abstract and §1] BFGS is described as a 'momentum-based optimizer.' BFGS is a quasi-Newton method, not a momentum-based method, and this terminology should be corrected.
  5. [§4] The claim that convergence 'almost doubles' should be quantified with a specific threshold and the underlying accuracy values, e.g., 'at threshold e=0.05, accuracy increases from A% to B%.'
  6. [Figure 2] The subcaptions in Figure 2 are garbled (e.g., '(y0), '(y0) /30'); they should be cleaned up for readability.
  7. [Appendix A, Table 1] The 'Sampling Rate' column is used without a definition; it should be explained how often control parameters are sampled per trajectory in each setup.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline convergence gain is reported for the 'top-performing' regularized ProxyNN with hyperparameters swept per benchmark, so the main improvement is selected on test data rather than predicted from a fixed principle.

  1. fitted input called prediction [Section 3.3 ('Optimization Performance Evaluation'), Table 1, Figures 4-5]
    "We further report the convergence accuracy for the top-performing regularized proxy network for all inverse setups discussed in (§ 3.1) for a range of prediction error thresholds. ... Table 1: Billiards-2D ... ({0.013, 0.018, 0.025, 0.027, 0.050}, 5)."

    The reported accuracy is for the network that performs best on the same benchmark used to demonstrate the method. Table 1 lists swept ranges of (sigma, mu) per system, and no validation split or model-selection criterion is described anywhere in the paper. The central empirical claim—'the convergence to the optimal solution almost doubles when using BFGS on ProxyNN predicted loss compared to ground truth loss in the 2-D billiards setup' (Section 4)—is therefore a post-selection statistic: the regularization hyperparameters are fitted to the test outcomes that the claim is then used to explain.

full rationale

The paper has no self-citation chain doing load-bearing work: PhiFlow (Holl et al., 2020) is cited only as the simulation framework, and the regularization motivation (Dherin et al., 2022) involves no overlapping authors. The two-step optimization (primary BFGS on the proxy, secondary BFGS on the ground-truth loss) is an empirical proposal whose key assumption—that regularized proxies preserve the basin of the true global minimum—is not derived anywhere; Figure 4 explicitly shows that at sigma = 0.05 the basin is 'too simplified' and gains vanish. That is an unverified assumption and a correctness risk, but not a circularity. The one concrete circular step is the evaluation: Section 3.3 reports only the 'top-performing regularized proxy network' out of a swept set of hyperparameters, with no validation split, so the headline 'convergence almost doubles' is a best-case selection on the test benchmarks. I do not count the Eq. (4) inconsistency (the prose describes a penalty only where f_theta > L, while Eq. (4) scales the entire loss by mu) as circularity; it is an internal correctness issue. Score 6 reflects partial circularity: the reported improvement is partly forced by test-set selection, though the smoothing mechanism itself has independent empirical content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central method depends on the surrogate approximation, the accuracy of the forward solver, and the untested assumption that regularization preserves the global minimum. No new physical entities are introduced. The tuning of sigma and mu on the test problems is the main free-parameter concern.

free parameters (2)
  • sigma (input noise scaling) = e.g., 0.0125, 0.025, 0.05, 0.013, 0.018, 0.021, 0.1 across problems
    Gaussian noise is added to sampled control parameters during training, with sigma selected per problem to maximize test convergence accuracy (Section 2.3, Table 1, Figure 4).
  • mu (loss-penalty multiplier) = 1 (no penalty) or 5 (with penalty), per Table 1 and Figure 5
    When f_theta > L, the training loss is multiplied by mu > 1 to bias predictions toward lower loss values; mu is chosen per problem together with sigma.
assumptions (4)
  • standard math Deep neural networks can approximate the configuration loss L with sufficient accuracy using the given architectures and training data.
    The method assumes universal approximation and generalization of CNNs on these data sets (Section 2.2).
  • domain assumption The numerical solver P produces accurate ground-truth trajectories and loss values L for training and evaluation.
    All training labels and final BFGS refinements rely on P(Y0, Xs) computed by simulation (Section 2.1).
  • domain assumption The configuration loss L has a global minimum at X* and is suitable for gradient-based refinement near that minimum.
    The inverse problem is defined by minimizing L, and the secondary BFGS step assumes a smooth basin of attraction at the optimum (Section 2.1, Section 2.4).
  • ad hoc to paper Noise and loss-penalty regularization flatten the proxy landscape while preserving the location and basin of the global minimum.
    This is the core heuristic that makes the method work; it is not proven and is shown to fail for high sigma in Figure 4.

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Cite this review

Pith. "Pith review of Optimization Landscapes Learned: Proxy Networks Boost Convergence in Physics-based Inverse Problems." pith.science (2026). https://pith.science/paper/BDWV4BOE

@misc{pith2026250116573,
  author       = {Pith},
  title        = {Pith review of: Optimization Landscapes Learned: Proxy Networks Boost Convergence in Physics-based Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDWV4BOE}},
  note         = {Machine review of arXiv:2501.16573}
}
read the original abstract

Solving inverse problems in physics is central to understanding complex systems and advancing technologies in various fields. Iterative optimization algorithms, commonly used to solve these problems, often encounter local minima, chaos, or regions with zero gradients. This is due to their overreliance on local information and highly chaotic inverse loss landscapes governed by underlying partial differential equations (PDEs). In this work, we show that deep neural networks successfully replicate such complex loss landscapes through spatio-temporal trajectory inputs. They also offer the potential to control the underlying complexity of these chaotic loss landscapes during training through various regularization methods. We show that optimizing on network-smoothened loss landscapes leads to improved convergence in predicting optimum inverse parameters over conventional momentum-based optimizers such as BFGS on multiple challenging problems.

Figures

Figures reproduced from arXiv: 2501.16573 by the authors.

Figure 1
Figure 1. Schematic representation of formulating the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. System trajectories and configuration loss landscapes (L) for non-linear inverse problems in (a,e) Burgers equation, (b,f) for the Kuramoto-Sivashinsky equation, and (c,d) and (g,h) for the Billiards-2D setup respectively. These landscapes (f-h) encounter convergence issues in iterative optimizers due to local minima and regions of sharp and vanishing gradients. hyperbolic partial differential equation that describe… view at source ↗
Figure 3
Figure 3. ProxyNNs predict configuration loss landscapes L. 0.5 0.6 0.7 0.8 y0 20 40 60 Unregularized Proxy NN 0.5 0.6 0.7 0.8 y0 20 40 60 = 0.012 Proxy NN 0.5 0.6 0.7 0.8 y0 20 40 60 = 0.025 Proxy NN 0.5 0.6 0.7 0.8 y0 20 40 60 = 0.050 Proxy NN 0.00 0.05 0.10 <e> threshold 0 10 20 30 40 % convergence BFGS GD NN 0.00 0.05 0.10 <e> threshold 10 20 30 40 % convergence Noise( = 0.012) BFGS GD NN 0.00 0.05 0.10 <e> threshold 20 3… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Predicted Loss Landscapes (top) and Optimization performance (bottom) of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: For 256 unique inverse problems for each setup, we report convergence accuracy with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: True Vs Predicted Values. The improvement in convergence accuracy us￾ing ProxyNN can be attributed to the smoothen￾ing of the loss via the regularization method and the two-step optimization process. The primary optimization step on the regularized ProxyNNs effectively…
Figure 7
Figure 7. Figure 7: Two-Step optimization policy for ProxyNNs. The primary step aims to converge in the vicinity of the global minimum which is treated as initial guess for the secondary optimization step. C ProxyNN predicted Configuration Loss landscapes for 4D Billiards 0.5 0.6 0.7 0.8 …
Figure 8
Figure 8. Figure 8: Proxy networks predicts configuration loss landscape L in the 4-D Billiards Inverse Setup. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.