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REVIEW 4 major objections 5 minor 27 references

A condensing approach to multiple shooting neural ordinary differential equation

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By condensing multiple-shooting continuity constraints into a linear solve, the paper makes it possible to train neural ODEs with Adam and shows this beats single-shooting on several oscillatory systems.

desk verdict Plausible, practically motivated adaptation of condensing to MS-NODE training, but the printed algorithm has a bug and the paper overclaims the constraint guarantee under Adam. read the letter →

arxiv 2506.00724 v1 pith:GHPFVXNN submitted 2025-05-31 cs.LG math.DS

Siddharth Prabhu , Srinivas Rangarajan , Mayuresh Kothare This is my paper
classification cs.LGmath.DS
keywords multiple-shootingneuralordinarydifferentialequationsequalityconstraintscondensingautomaticdifferentiationAdamoptimizeroscillatorysystemsparameterestimation
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 aims to make multiple-shooting practical for training neural ordinary differential equations (NODEs). Multiple-shooting splits a trajectory into short intervals and enforces equality constraints that close the gaps between consecutive intervals; this is more stable than single-shooting on long oscillatory systems, but the constraints are hard to enforce with first-order optimizers. The authors derive a condensing step that projects gradient updates onto the constraint manifold by solving a linear system built from Jacobian-vector products, then feed the resulting updates to Adam. They demonstrate the procedure on a suite of oscillatory test problems, reporting that the trained models match the underlying dynamics on held-out test data while single-shooting fails on the same problems.

What carries the argument

The central object is the condensed KKT update obtained from the multiple-shooting Lagrangian $L(x,p,\lambda)=\Phi+\lambda^\top G$, where $G$ is the vector of shooting-gap constraints $x_{k+1}-F_k(x_k,p)$. Replacing the Hessian block in the Newton system with identity blocks yields updates $\Delta x = -(G_x^\top \Delta\lambda + L_x)$ and $\Delta p = -(G_p^\top \Delta\lambda + L_p)$ with $\Delta\lambda = (G_x G_x^\top + G_p G_p^\top)^{-1}[G - G_x L_x - G_p L_p]$. The paper computes this update in two ways: for the large-parameter setting it uses conjugate gradient with Jacobian-vector and vector-Jacobian products so that the matrix $G_x G_x^\top + G_p G_p^\top$ never has to be formed, and for small problems it assembles $G_x$ directly by exploiting its block-bidiagonal sparsity. The resulting $\Delta p$ is fed to Adam, so the optimizer sees a constraint-corrected direction at each step.

What would settle it

A concrete test: run the released code on the Lotka-Volterra benchmark with the reported hyperparameters, but initialize each shooting interval from a slightly perturbed state (say 10% noise added to the preceding measurement). If the norm of the shooting gap does not decrease to the reported $10^{-4}$ level, or if the conjugate-gradient solve stalls because $G_x G_x^\top + G_p G_p^\top$ is singular at a feasible point, the claim that first-order training can satisfy the shooting constraints fails.

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Extended reading notes

Core claim

The paper's central claim is that the equality-constrained multiple-shooting training problem can be solved by a first-order optimizer. Replacing the Hessian in the Newton KKT system with identity blocks produces condensed updates that satisfy the first-order Taylor expansion of the shooting constraints at each iteration, and these updates can be computed cheaply with conjugate gradient and automatic-differentiation products. The empirical claim is that MS-NODE training reproduces the training trajectory and extrapolates to unseen test dynamics for systems such as Lotka-Volterra, Goodwin, Van der Pol, FitzHugh-Nagumo, Brusselator, and Oregonator, while a single-shooting baseline does not. The paper also reports cases (KM, MHD, Calcium Ion) where the method fits the training data but fails to generalize, interpreting this as overfitting rather than a failure of the constraint mechanism.

Load-bearing premise

The method depends on the assumption that replacing the full Newton Hessian in the KKT system with identity matrices yields a step that Adam can follow to a point satisfying the shooting constraints, and that the matrix $G_x G_x^\top + G_p G_p^\top$ is invertible at every iteration; the paper gives no convergence proof and does not discuss regularization.

Editorial extensions

If this is right

  • Multiple-shooting training for NODEs no longer requires a second-order or augmented-Lagrangian solver; the condensing step is compatible with a standard Adam training loop.
  • Because the reported constraint violation $|G|_\infty$ converges to the order of $10^{-4}$, the trained model can be simulated as a single trajectory at test time without leaving a large shooting gap.
  • The benchmarks show the method handles long oscillatory trajectories where single-shooting is reported to diverge, including the Lotka-Volterra, Van der Pol, and Oregonator systems.
  • For delayed and stiff systems such as KM and MHD, the constraint solver still fits the training data, so the method provides a continuous surrogate even when the learned dynamics do not extrapolate.
  • The machinery is not tied to a particular integration scheme: the discretize-then-optimize sensitivities only require forward and reverse automatic differentiation through the integrator.

Reading between the lines

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

  • A natural extension not tested in the paper is to feed the condensed update to other first-order optimizers or to apply the same projection blockwise in mini-batch training; the derivation does not depend on Adam specifically.
  • The paper does not regularize the linear solve, and if $G_x G_x^\top + G_p G_p^\top$ becomes singular or ill-conditioned (for instance, when shooting intervals are nearly redundant), the update is undefined; adding Tikhonov regularization would be a practical safeguard.
  • Because the condensing step only requires sensitivities of the constraints, it could also be applied to parameter estimation in known ODE models or to trajectory optimization, where it would avoid assembling and inverting full Hessians.
  • The observed generalization failures on KM, MHD, and Calcium Ion suggest that once the constraints are satisfied, the remaining bottleneck is model capacity or identifiability; a testable follow-up would be to check whether the generalization gap closes with early stopping or wider networks.
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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 / 5 minor

Summary. The paper proposes a condensing-based method for training multiple-shooting neural ODEs (MS-NODE) with first-order optimizers such as Adam. The authors formulate the KKT conditions for the shooting-constrained problem, replace the Hessian with an identity approximation, and derive a linear system for updates to shooting states, parameters, and Lagrange multipliers (Eqs. 5-10). They then present algorithms for Jacobian-vector products exploiting the sparsity of the constraint Jacobian, and use conjugate gradient to solve the condensed system. Experiments on ten oscillatory dynamical systems compare training and test MSE against single-shooting, with three systems (MHD, KM, Calcium Ion) failing to generalize.

Significance. If it worked as claimed, a first-order-compatible condensing approach to multiple-shooting NODEs could substantially improve training stability for stiff and oscillatory systems, a problem of practical importance. The paper also provides a useful sparsity-exploiting formulation for Jacobian-vector products and makes code available. However, the manuscript contains multiple internal inconsistencies between the derivation and the implemented algorithms, and the coupling of the condensing step with Adam is not justified. These issues currently prevent the central claim from being supported.

major comments (4)
  1. [Section 3 (Algorithm 5)] Algorithm 5 swaps the assignments of Δx and Δp relative to Eqs. (7)-(8). Eq. (7) gives Δx = -(Gx^T Δλ + Lx) and Eq. (8) gives Δp = -(Gp^T Δλ + Lp), but Algorithm 5 returns Δp = -Lx - ALGORITHM4(Δλ) and Δx = -Lp - ALGORITHM2(Δλ). This is not a notational slip: a reader implementing the printed algorithm will compute different directions, and the derived constraint-satisfaction property GxΔx + GpΔp = -G no longer holds for those outputs.
  2. [Section 2 (Algorithms 1, 2, 4)] The Jacobian-vector product algorithms are sign-inconsistent with the definitions of Gx and Gp. In Eq. (12), the nonzero blocks of Gp are -∂F_{k-1}/∂p, yet Algorithm 1 sets Y[k] = ∂F_{k-1}/∂p v and Algorithm 2 accumulates V[k]^T ∂F_{k-1}/∂p, both with no minus sign. Similarly, Algorithm 4 initializes Y[m] = -V[m] and then uses Y[k-1] = V[k-1] - Y[k]^T ∂F_{k-1}/∂x_{k-1}; for m=2 this yields (Gx^T v)_1 = v_1 + v_2^T ∂F, whereas Eq. (11) gives (Gx^T v)_1 = v_1 - v_2^T ∂F. These errors propagate into the HVP used to compute Δλ and into the final updates.
  3. [Section 3, first paragraph] The paper states that the updates to x and p satisfy the first-order Taylor expansion of the equality constraints at each iteration, but then feeds the resulting Δx and Δp into the Adam optimizer. Adam applies a different positive rescaling to every coordinate based on its moment estimates, so the actual update is (Dx Δx, Dp Δp) with diagonal Dx, Dp that are not scalar multiples of the identity. Unless Dx and Dp are both scalar, Gx(Dx Δx) + Gp(Dp Δp) + G is generally nonzero. The manuscript gives no explanation of how the condensing guarantee survives Adam's rescaling, and the final |G|∞ values in Table 1 cannot establish a per-iteration property. As written, the central mechanism of the paper does not operate under the proposed optimizer.
  4. [Sections 3.8-3.10 and Table 1] The empirical claim is weakened by the fact that three of ten systems (MHD, KM, Calcium Ion) do not generalize to test data, with test loss reported as "-". The paper acknowledges this, but the abstract and introduction claim that multiple shooting gives better generalization than single shooting on oscillatory systems without stating the boundary of this claim. In addition, no comparison is made against existing multiple-shooting NODE training methods (e.g., Turan and Jäschke 2022 or Massaroli et al. 2021) on the same benchmarks, and the train/test protocol (how unseen test data were generated, how scaling was applied, how hyperparameters were selected) is not fully described, which limits reproducibility.
minor comments (5)
  1. [Equation (1)] The subscript in the integral limit "tK" appears to be a typo for "tk".
  2. [Equation (2)] The text says "G is a matrix with m constraints," but G is a column vector of dimension mn×1; the wording should be corrected.
  3. [Section 2 (Algorithm 2)] In Algorithm 2, the update "y←y+V[k]^T ∂F_{k-1}/∂p" mixes a p×1 vector with what would be a 1×p row vector; the intended operation is likely y←y+(∂F_{k-1}/∂p)^T V[k].
  4. [Section 3, first paragraph] The text contains a typo: "The code is availabe" should be "The code is available".
  5. [Section 2.1 (Figure 2 caption)] The caption says the Jacobian Gx can be computed in m JVP calls, but the surrounding text correctly says the number of required JVPs is proportional to n (the state dimension); the caption should be harmonized with the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the MS-NODE condensing derivation is self-contained and its reported test losses are genuinely out-of-sample.

full rationale

The claimed derivation runs from the KKT conditions (Eq. 4) to the condensed step (Eqs. 7-10) by algebraically solving a Gauss-Newton-like system (Eq. 6); the shooting constraints are the defining equations of the multiple-shooting problem, not quantities fitted from data whose prediction is then reported. The train/test comparison in Section 3 is out-of-sample: the Table 1 test losses are MSE on unseen trajectories, and no parameter is fitted to that test data and then renamed a prediction. The cited prior work (multiple shooting, condensing, Adam) is background and is not used as an unverified self-citation chain; the reference to Rangarajan et al. (2022) concerns linear equality constraints and is not load-bearing for the condensing derivation. The paper does contain non-circular technical gaps: Section 3 applies Adam to the condensed direction without proving that Adam's per-coordinate rescaling preserves the linearized constraint, and Algorithm 5's assignments for Delta p and Delta x appear swapped relative to Eqs. 7-8. These are correctness or reproducibility concerns, not circularity, because the output is not equivalent to the input by construction. The acknowledged generalization failures in Section 4 are stated limitations, not hidden fitted predictions. Therefore no significant circularity is present.

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

The method introduces no new physical entities or fitted physical constants. The free parameters are experimental hyperparameters (intervals, learning rates, architecture) that the authors chose by hand. The main unstated assumptions are about the validity of the identity-Hessian step, the invertibility of the linear system, and the adequacy of the integration and sensitivity schemes.

free parameters (4)
  • number of shooting intervals m = 20 to 100 per system (Table 1)
    Chosen by hand per test system; affects conditioning of the constraint Jacobian and the optimization landscape.
  • learning rate and schedule = 0.01 with decay (Table 1)
    Hand-tuned per system; no systematic selection or sensitivity analysis.
  • neural network architecture (hidden units per state) = e.g., [32, 64, 32] for Lotka-Volterra (Table 1)
    Chosen by hand; impacts model expressiveness and is not derived from any criterion.
  • CG solver tolerance for Δλ = not reported
    The paper uses conjugate gradients for Eq. (9) but does not state tolerance, max iterations, or preconditioning.
assumptions (4)
  • ad hoc to paper The KKT system with identity Hessian approximation (Eq. 6) yields a valid descent step for first-order training.
    No proof is given; Eq. (6) replaces the Hessian with identity without justification, and the resulting step is fed to Adam as a gradient.
  • domain assumption GxGx^T + GpGp^T is nonsingular and CG converges.
    Invertibility is assumed by Eq. (9); no rank analysis or regularization is discussed.
  • domain assumption Discretize-then-optimize sensitivities match the numerical integration scheme.
    Sensitivities are computed with forward and reverse AD across F (Section 2); no solver or tolerance is specified.
  • domain assumption A separate neural network per state variable can represent the vector field of each system.
    Architecture in Figure 3; no expressiveness argument tied to the specific systems tested.

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Cite this review

Pith. "Pith review of A condensing approach to multiple shooting neural ordinary differential equation." pith.science (2026). https://pith.science/paper/GHPFVXNN

@misc{pith2026250600724,
  author       = {Pith},
  title        = {Pith review of: A condensing approach to multiple shooting neural ordinary differential equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHPFVXNN}},
  note         = {Machine review of arXiv:2506.00724}
}
read the original abstract

Multiple-shooting is a parameter estimation approach for ordinary differential equations. In this approach, the trajectory is broken into small intervals, each of which can be integrated independently. Equality constraints are then applied to eliminate the shooting gap between the end of the previous trajectory and the start of the next trajectory. Unlike single-shooting, multiple-shooting is more stable, especially for highly oscillatory and long trajectories. In the context of neural ordinary differential equations, multiple-shooting is not widely used due to the challenge of incorporating general equality constraints. In this work, we propose a condensing-based approach to incorporate these shooting equality constraints while training a multiple-shooting neural ordinary differential equation (MS-NODE) using first-order optimization methods such as Adam.

Figures

Figures reproduced from arXiv: 2506.00724 by the authors.

Figure 1
Figure 1. A pictorial representation of single-shooting (a) and multiple-shooting (b) in the context of parameter [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The Jacobian Gx can be computed in only m JVP calls by exploiting its sparsity structure, compared to mn JVP calls required by naive approach Unfortunately, the Jacobian Gp, defined in equation 12, is dense, so its sparsity structure cannot be exploited. However, in the case of neural networks, where p ≫ mn, we compute the Jacobian using mn VJP’s instead of p JVP’s [Griewank and Walther, 2008]. Gp =         … view at source ↗
Figure 3
Figure 3. A pictorial representation of the neural network architecture used for training multiple-shooting neural [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: Comparison between the model trained using multiple-shooting (red) and the model trained using single [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Performance of the model trained using multiple-shooting (red) on training (blue) generated from the KM [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Performance of the model trained using multiple-shooting (red) on training (blue) generated from the [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

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