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REVIEW 2 major objections 6 minor 35 references

Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper derives a Hessian-vector product algorithm for implicit finite-element differentiation and shows that exact Hessians rescue Newton-CG where finite-difference Hessians diverge.

desk verdict A correct and useful Hessian-vector product primitive for implicit FEM, with benchmark claims that overreach because the finite-difference comparison is uncontrolled. read the letter →

arxiv 2505.12646 v1 pith:U2C7UPJX submitted 2025-05-19 cs.CE

Tianju Xue This is my paper · ORCID
classification cs.CE
keywords implicitdifferentiationHessian-vectorproductPDE-constrainedoptimizationdifferentiablephysicsfiniteelementmethodautomaticNewton-CGinverseproblems
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

This paper addresses a gap in differentiable physics: first-order gradients of implicit finite-element solvers are standard, but second-derivative (Hessian) information has been missing. It derives and implements a Hessian-vector product for PDE-constrained optimization where the state is defined implicitly by a residual equation $r(y,\theta)=0$, using only primitive automatic-differentiation operations. Accuracy is verified by comparison with finite differences and by a Taylor remainder test showing third-order convergence of the second-order expansion. On four benchmarks, Newton-CG with the exact Hessian converges reliably for nonlinear inverse problems, while a finite-difference Hessian version fails on two of them.

What carries the argument

The load-bearing object is the bordered system of Eq. (15), formed from the Lagrangian $L(y,\lambda,\theta)=g(y,\theta)+\lambda^T r(y,\theta)$; its solution gives $\partial y/\partial\theta$ and $\partial\lambda/\partial\theta$. Algorithm 1 rewrites that system as two linear solves for the incremental state $\hat{y}$ and incremental adjoint $\hat{\lambda}$, then evaluates Eq. (22) to produce $H\hat{\theta}$. The essential mechanism is that second-order terms such as $(\partial/\partial\theta)(\lambda^T\partial r/\partial y)\hat{\theta}$ can be interpreted as nested Jacobian-vector and vector-Jacobian products, making the Hessian available solely from AD primitives rather than from hand-derived formulas.

What would settle it

Run the two benchmarks where Newton-CG (FD) diverges (source-field identification and traction identification) with a systematically tuned finite-difference step size, including central differences over a range of $h$ and gradient scaling. If a tuned finite-difference Hessian-vector product converges in comparable time, the paper's central contrast between exact and approximate Hessians collapses.

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Extended reading notes

Core claim

The central claim is that the Hessian $d^2g/d\theta^2$ of an objective subject to an implicit finite-element constraint can be computed exactly, direction by direction, without forming the full matrix. The Hessian-vector product $H\hat{\theta}$ is assembled from four solves\textemdash forward state, adjoint, incremental forward, and incremental adjoint\textemdash combined in Eq. (22); each solve is linear even when the forward problem is nonlinear. All second-order terms in the assembly are compositions of Jacobian-vector and vector-Jacobian products, so a differentiable-programming library with those primitives can supply curvature automatically. The Taylor remainder test confirms that the residual of the second-order Taylor expansion decays as $\epsilon^3$, consistent with a machine-precision Hessian-vector product.

Load-bearing premise

The empirical conclusion that finite-difference Hessians are unusable rests on the assumption that the observed failures are inherent to finite differences rather than artifacts of an untuned step size; the derivation also assumes the bordered system matrix $\partial r/\partial y$ is nonsingular at every evaluation point.

Editorial extensions

If this is right

  • On the nonlinear traction identification and shape-optimization benchmarks, Newton-CG with exact Hessian-vector products converges in wall-clock time faster than L-BFGS-B.
  • On the linear source-identification and thermal-mechanical control benchmarks, L-BFGS-B converges faster, so the extra curvature is not always worth its cost.
  • Finite-difference Hessian-vector products fail to converge in two of the four benchmarks, so exact implicit Hessians buy robustness, not just speed.
  • Because the implicit Hessian approach solves each incremental problem once per direction and can cache the state and adjoint, repeated Hessian-vector products at a fixed parameter point are cheaper than repeated finite-difference gradient pairs.

Reading between the lines

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

  • Beyond the paper: because the derivation only assumes an implicit residual equation, the same Hessian-vector product recipe transfers to non-finite-element implicit solvers.
  • Beyond the paper: exact curvature enables Hessian-based uncertainty quantification, such as Laplace approximations of posterior covariances for PDE-constrained inverse problems.
  • Beyond the paper: the profiling results suggest that the choice of AD composition mode matters, and that exploiting sparsity in the second-order terms is a natural next step toward much larger problem sizes.
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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

2 major / 6 minor

Summary. The manuscript derives an implicit Hessian-vector product for PDE-constrained optimization in a discretize-then-optimize framework, implements it in JAX-FEM using JVP/VJP primitives, and validates the implementation against finite-difference approximations and a Taylor remainder test. Four benchmark inverse problems are used to compare L-BFGS-B, Newton-CG with finite-difference Hessian-vector products (Newton-CG (FD)), and Newton-CG with the proposed implicit-differentiation Hessian-vector products (Newton-CG (AD)). The paper concludes that Newton-CG (AD) is robust and accelerates convergence for nonlinear problems, while L-BFGS-B is sufficient for linear problems, and that finite-difference Hessian approximations are unstable and inaccurate.

Significance. The core methodological contribution is sound and useful: the derivation in Section 2.3 is algebraically consistent, the algorithm in Section 3.1 is a natural and efficient composition of first-order implicit differentiation with JVP/VJP primitives, and the Taylor remainder test in Figure 5 provides independent, quantitative evidence that the Hessian-vector product is computed correctly to the expected third-order convergence. The public code is a strength, and the paper fills a real gap in differentiable FEM by making curvature information available as a primitive. The practical benchmark claims are only partially supported, however, because the comparison with finite-difference Hessians is not controlled and several benchmark settings are unreported. With those gaps fixed, the paper would make a solid contribution to the differentiable-physics community.

major comments (2)
  1. [Sections 4.1, 4.2, and 4.5] The claim that exact Hessians avoid "the instability and inaccuracy of finite difference Hessian approximations" is not established by the reported experiments, because Newton-CG (FD) is run with SciPy's default finite-difference Hessian-vector product and no step size, gradient tolerance, or other tuning is reported. The authors' own validation in Section 3.3.1 (Figures 3 and 4) shows that the finite-difference Hessian-vector product is strongly step-size dependent: the relative difference ranges from about 1e-3 at h=0.1 to about 1e-7 at h=1e-4. SciPy's default step for Newton-CG without a supplied Hessian is typically of order sqrt(machine epsilon) ~ 1.5e-8, which is two orders of magnitude smaller than the smallest h validated here and far below the usual optimum for a second-derivative estimate. The non-convergence of Newton-CG (FD) in Sections 4.1 and 4.2 may therefore be an artifact of the default step rather than of finite-difference Hessians in general. Please add a tuned-FD comparison, for example with h in the range 1e-4 to 1e-3 using Equation (28), and report all optimizer parameters; if tuned FD still fails to converge, the robustness claim is supported.
  2. [Section 4 (Eqs. (39), (43), (48), (49))] The benchmark protocol is under-specified and not fully reproducible: the regularization weights α introduced in Equations (39), (43), and (48) are never assigned numerical values, the optimizer tolerances, initial guesses, and linear solver settings are not reported, all convergence histories come from single runs, and the shape-optimization design mapping from the rotation angles to the density field in Section 4.4 is not defined. Consequently, the quantitative timing comparisons and the Section 4.5 recommendations ("Newton-CG faster for nonlinear, L-BFGS-B for linear") are not yet supported by the evidence as presented. Please report the complete settings for each benchmark and, where feasible, include multiple restarts or perturbation studies to show that the qualitative conclusions are robust.
minor comments (6)
  1. [Section 4 (multiple places)] The text repeatedly says "PDF-constrained optimization problem" where it should say "PDE-constrained optimization problem."
  2. [Figures 3 and 4] The x-axis tick labels are corrupted in the typeset version (for example, "0⊿000960" instead of a numeric value with a power-of-ten notation); the figures should be regenerated with proper mathematical notation.
  3. [Section 3.3.1] The step-size sweep stops at h=10^-4; adding values near the expected optimum for a second-derivative estimate, roughly h ~ eps^(1/3), would make the accuracy validation more convincing.
  4. [Section 2.3, Eq. (15)] The derivation implicitly assumes that the bordered KKT matrix, in particular ∂r/∂y, is nonsingular at every evaluation point; this implicit-function-theorem hypothesis should be stated explicitly as an assumption on the admissible parameter set.
  5. [Section 4.4] The mapping from the rotation angles θ to the density field ρ(x) through the sigmoid function is not defined; for reproducibility, the manuscript should give the exact formula and any smoothing or length-scale parameters.
  6. [Algorithm 1] Steps 1 and 2 require solving a nonlinear forward problem and a linear adjoint problem, respectively; the intended numerical solver and convergence criterion for these solves should be specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the implicit Hessian-vector product is derived from the residual and objective via the Lagrangian, and the Taylor remainder test independently validates it against objective evaluations.

full rationale

The paper's central object, the implicit Hessian-vector product Hθhat, is derived in Section 2.3 by differentiating the first-order optimality conditions of the Lagrangian (Eqs. (15)-(22)); this is a self-contained derivation from the residual r(y,theta)=0 and objective g(y,theta), not an input fitted to the outputs it claims to predict. The implementation in Algorithm 1 composes JAX JVP/VJP primitives to evaluate those derived terms, and accuracy is checked two ways: against central finite differences of the derived gradient (Eq. (28)) and, independently, against the third-order Taylor remainder (Eq. (36), Fig. 5). Neither check reuses the Hessian formula as its own evidence: the Taylor test compares the computed gradient and Hessian against direct objective evaluations, and the FD comparison uses a different computation path. The only self-citation to JAX-FEM [13] supplies the first-order gradient baseline and FEM infrastructure, but the gradient is also derived in Eqs. (12)-(14), and the Hessian derivation does not assume the truth of that citation. The benchmarking claim that Newton-CG (FD) is unreliable because SciPy's default FD Hessian failed in two examples is an experimental-control concern, not circularity, since the comparison is not constructed from fitted values or definitions that guarantee the conclusion. No self-definitional, fitted-input, or citation-imported uniqueness step was found.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central derivation assumes standard implicit-function and differentiability conditions on the discrete residual and objective, and the benchmark conclusions additionally assume the FD optimizer baseline is set up fairly. No new physical entities are introduced; the only user-chosen free parameter is the regularization weight α, which is not reported.

free parameters (1)
  • Regularization weight α in benchmarks = Not reported
    Defined in Eqs. (9), (39), (43), (48) but no numerical value is given; the convergence speeds and final objectives of Newton-CG versus L-BFGS-B can depend on this trade-off parameter.
assumptions (3)
  • domain assumption The discrete residual r(y,θ)=0 implicitly defines y(θ) and the bordered KKT matrix in Eq. (15) is nonsingular (in particular ∂r/∂y is invertible).
    Required for the adjoint, incremental forward, and incremental adjoint solves in Algorithm 1 to be well-posed; nowhere stated or checked.
  • standard math The Lagrangian L is C² so mixed partials commute, as used to equate the transposed block vector in Eq. (19).
    Standard Schwarz theorem; the paper relies on it silently.
  • domain assumption JAX's JVP and VJP primitives compute exact derivatives of the assembled finite element residual and objective.
    The entire implementation in Section 3.2 is built on this contract; correctness depends on it.

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Cite this review

Pith. "Pith review of Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics." pith.science (2026). https://pith.science/paper/U2C7UPJX

@misc{pith2026250512646,
  author       = {Pith},
  title        = {Pith review of: Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2C7UPJX}},
  note         = {Machine review of arXiv:2505.12646}
}
read the original abstract

Differentiable programming is revolutionizing computational science by enabling automatic differentiation (AD) of numerical simulations. While first-order gradients are well-established, second-order derivatives (Hessians) for implicit functions in finite-element-based differentiable physics remain underexplored. This work bridges this gap by deriving and implementing a framework for implicit Hessian computation in PDE-constrained optimization problems. We leverage primitive AD tools (Jacobian-vector product/vector-Jacobian product) to build an algorithm for Hessian-vector products and validate the accuracy against finite difference approximations. Four benchmarks spanning linear/nonlinear, 2D/3D, and single/coupled-variable problems demonstrate the utility of second-order information. Results show that the Newton-CG method with exact Hessians accelerates convergence for nonlinear inverse problems (e.g., traction force identification, shape optimization), while the L-BFGS-B method suffices for linear cases. Our work provides a robust foundation for integrating second-order implicit differentiation into differentiable physics engines, enabling faster and more reliable optimization.

Figures

Figures reproduced from arXiv: 2505.12646 by the authors.

Figure 1
Figure 1. Differentiable programming breaks the boundary between deep learning and differentiable [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The forward prediction part of the model problem for implicit differentiation. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Relative difference ev of Hessian-vector product evaluations between the proposed implicit differentiation approach and the finite difference approach. In each subfigure, 100 samples are used to generate the histogram plot. As shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Relative difference es of Hessian-vector product evaluations between the proposed implicit differentiation approach and the finite difference approach. In each subfigure, 100 samples are used to generate the histogram plot. As shown in [PITH_FULL_IMAGE:figures/full_fi…
Figure 5
Figure 5. Figure 5: Taylor remainder test. As expected, the zeroth-order expansion of the residual achieves [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Execution time for the three ways of AD mode composition: “fwd-rev” stands for “forward [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The forward prediction of the linear Poisson’s problem. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Optimization results for the source field identification problem. The source fields at [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Matching between the predicted solution fields (colored plot) and the observed solution [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: The forward prediction of the hyperelasticity problem. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Optimization results for the traction field identification problem. In (a), the objective [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Matching between the predicted displacement fields (colored plot) and the observed [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: The forward prediction of the thermal-mechanical problem. The deformation is scaled [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Optimization results for the thermal-mechanical control problem. In (a), the objective [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: The deformed object at optimization steps (I) to (IV), with corner displacement gradually [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: The forward prediction of the shape optimization problem. [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Optimization results for the shape optimization problem. The configurations of the [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]

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