Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations

Stanley Osher; Yesom Park

REVIEW 4 major objections 6 minor 76 references

Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A single implicit formula, learned by a neural network, solves high-dimensional Hamilton–Jacobi equations without grids or Legendre transforms.

desk verdict Clean idea and promising numerics, but the central uniqueness theorem is unproven and Example 2.1 contains a concrete formula error. read the letter →

arxiv 2501.19351 v1 pith:42LGUJF6 submitted 2025-01-31 cs.LG

Yesom Park , Stanley Osher This is my paper · ORCID

classification cs.LG MSC 65M2568T0735C99

keywords Hamilton-Jacobiequationsimplicitsolutionformulaviscositysolutionsdeeplearningmethodofcharacteristicshigh-dimensionalPDEscursedimensionalityoptimalcontrol

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 claims that the viscosity solution of a Hamilton-Jacobi equation can be written through an implicit equation that involves the solution's own gradient at the same point, and that this equation can be turned into a deep-learning objective. The formula is derived from the method of characteristics, and the authors prove that, when the Hamiltonian or the initial function is convex, a continuous function satisfying it coincides with the viscosity solution almost everywhere. Because the formula avoids Legendre transforms and explicit characteristic trajectories, it allows a mesh-free neural network to solve HJ equations in tens of dimensions with training cost that grows little with dimension. The same idea is extended to state-dependent Hamiltonians by approximating characteristics piecewise linearly and marching in time. If correct, the method offers a practical route around the curse of dimensionality for HJ equations arising in control, level-set evolution, and geometric optics.

What carries the argument

The load-bearing object is the implicit solution formula (2.3), which packages the constant-gradient property of characteristics into a fixed-point condition on $u$ itself. It is derived from Hamilton's system $\dot x=\nabla H(p)$, $\dot u=-H+p^\top\nabla H(p)$, with $p=\nabla u(x,t)$ evaluated at the terminal point, and it substitutes $x_0=x-t\nabla H(\nabla u)$ into the initial data. The second piece of machinery is the neural representation: a multilayer perceptron with softplus activations, trained by gradient descent on a Monte Carlo estimate of the residual squared, with automatic differentiation supplying exact gradients. For state-dependent Hamiltonians, the machinery is a piecewise-linear approximation of characteristic curves over intervals of length $\Delta t$, which makes the implicit formula applicable step by step at first-order accuracy.

What would settle it

Find two different continuous functions on the same bounded spacetime domain that both satisfy (2.3) with the same initial condition; alternatively, take one of the nonconvex examples, compute the trained network's output, and compare it against a provably convergent monotone scheme at a fine resolution -- a mismatch would disprove the claim that the formula selects the viscosity solution.

Watch

Extended reading notes

Core claim

The paper's central object is the identity $$u(x,t)=-tH(\nabla u)+t\nabla u^\top\nabla H(\nabla u)+g(x-t\nabla H(\nabla u)),$$ which it calls the implicit solution formula. Along any characteristic the gradient $p=\nabla u$ is constant, so tracing a characteristic backward from $(x,t)$ reaches the initial point $x_0=x-t\nabla H(\nabla u)$, and the solution value is determined by the Hamiltonian and the initial function evaluated there. The authors show Theorems 2.1 and 2.2: under convexity (or concavity) of $H$ or of $g$, any continuous function satisfying this identity is the viscosity solution of the HJ equation almost everywhere, with the proof working by showing the identity is equivalent to the Hopf-Lax or Hopf formulas. The numerical method minimizes the squared residual of this identity with Monte Carlo samples, using a Lipschitz neural network to represent $u$, so no grid, no Legendre transform, and no characteristic ODE integration are needed. For state-dependent Hamiltonians the characteristic is approximated as linear over short time steps, yielding a first-order time-marching scheme.

Load-bearing premise

The proof assumes the implicit equation (2.3) has a unique continuous solution on the whole spacetime domain, since the theorems only show the viscosity solution satisfies the formula, not that every continuous solution of the formula must be that solution.

Editorial extensions

If this is right

HJ equations in 10 to 40 dimensions can be solved with a fixed network size, with per-epoch time and memory that barely change as dimension grows.
The method eliminates the need to compute Legendre transforms or to solve characteristic ODEs, so it applies to Hamiltonians and initial data where Hopf or Lax formulas are impractical.
Because the implicit formula encodes the initial condition automatically, training uses a single loss term rather than a tuned PDE/initial-condition weighting.
For state-dependent Hamiltonians the time-marching version offers a first-order convergent scheme whose error scales linearly with the time step $\Delta t$.
Nonconvex Hamiltonians, for which the theory is not yet established, are solved accurately in the reported experiments, suggesting the formula may extend beyond the proven convex cases.

Reading between the lines

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

If uniqueness of continuous solutions of (2.3) fails on bounded domains without boundary conditions, the training loss could select a spurious function; checking uniqueness or adding boundary data would close the gap the authors leave open.
The a.e. nature of the theorems means kinks are permitted; a natural test is whether the loss landscape has spurious minimizers exactly at the points where viscosity solutions lose differentiability.
The same fixed-point formula could be adapted to other first-order PDEs with constant gradient along characteristics, such as certain conservation laws or control problems, with the same mesh-free benefit.
A direct comparison of the trained network against a provably convergent monotone finite-difference solver on the nonconvex examples would test whether the empirical success extends where the proof does not.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing you should know: the implicit formula is a nice rearrangement of the characteristic equations, and turning it into a neural residual loss is a genuine idea. The numerics look encouraging, and the authors openly say the theory is incomplete. But the central theorems only prove one direction — that the viscosity solution satisfies (2.3) — and never prove the converse or uniqueness. That is the load-bearing piece, and it is missing.

The derivation in Section 2.1 is clean: from p constant along straight characteristics, you get exactly (2.3). The control-theoretic discussion in Section 2.2 connecting to PMP and Bellman is also coherent and useful. The experiments are extensive: smooth and nonconvex Hamiltonians, state-dependent cases, up to 40 dimensions, with near-constant per-epoch time. If the method works in practice at that scale, that is a useful contribution even without full theory.

Now the soft spots, in proportion. Theorems 2.1 and 2.2 assert that any continuous function satisfying (2.3) is the viscosity solution a.e. The proofs start from Hopf-Lax or Hopf and show the viscosity solution satisfies (2.3). They never establish that every continuous solution of (2.3) is the viscosity solution, or that such a solution is unique. Since (2.3) is a nonlinear first-order equation in x for each fixed t, you cannot just assume uniqueness; Example 2.1 is supposed to illustrate it, but the example has an actual error. For H(p)=p^2, (2.3) gives u = t(u_x)^2, not u = t u_x as written in (2.15). So the ODE analysis and the uniqueness argument in that example are irrelevant to the actual formula. That is a concrete mistake, not a matter of taste.

There is also a regularity issue: the theorem statements say 'continuous function', but (2.3) contains ∇u. A continuous function does not have a pointwise gradient in general, so the statement is not even well-posed without a differentiability assumption (e.g., Lipschitz or Sobolev). The authors sweep this under 'a.e.' without specifying the space.

On the numerical side, the paper gives no baselines, no error bars, no code, and no repeated-seed variability. The nonconvex and high-dimensional claims are empirical, and the authors admit the theory does not cover them. That is honest, but it means the abstract overstates what is established.

Bottom line: this is a promising numerical methods paper with a genuine gap between theorem and proof. It deserves a serious referee, not a desk reject, because the idea is new and the experiments are substantial — but it would need major revision to be publishable: fix Example 2.1, either prove uniqueness under explicit differentiability assumptions or state the converse as an open conjecture, and add proper numerical comparisons. I would bring it to reading group; I would not cite it yet.

Referee Report

4 major / 6 minor

Summary. The paper introduces an implicit solution formula (2.3) for Hamilton-Jacobi equations, derived from the method of characteristics by identifying the costate p with the current gradient ∇u. The authors claim in Theorems 2.1 and 2.2 that any continuous function satisfying (2.3) is the viscosity solution of (2.1) under convexity of either the Hamiltonian or the initial function, and they connect the formula to Hopf-Lax and Hopf representations. They then propose a deep learning algorithm that minimizes the residual of (2.3) via Monte Carlo sampling, avoiding grids and Legendre transforms, and extend the approach to state-dependent Hamiltonians through piecewise linear time marching. Numerical experiments include convex, concave, nonconvex, and state-dependent problems up to 40 dimensions.

Significance. If the equivalence between the implicit formula and the viscosity solution were rigorously established, the formula would be a genuinely useful alternative to Hopf-Lax/Hopf formulas, removing the need for Legendre transforms and characteristic ODE solving. The forward direction (the viscosity solution satisfies (2.3)) is a clean and potentially valuable observation, and the proposed mesh-free method is simple, scalable, and supported by a wide range of experiments. However, the central theorems as stated are not proven: they only show one direction, and the uniqueness of continuous solutions to (2.3) is asserted without proof. Moreover, Example 2.1, which is intended to illustrate uniqueness, contains a mis-derived formula for the quadratic Hamiltonian. These issues bear directly on the loss functional (3.1), since without uniqueness a neural minimizer is not guaranteed to recover the viscosity solution.

major comments (4)

[§2.1, Theorems 2.1 and 2.2] The theorems claim that any continuous function satisfying the implicit formula (2.3) is the viscosity solution a.e., but the proofs only demonstrate the forward direction: the viscosity solution, represented by the Hopf-Lax or Hopf formula, satisfies (2.3). No argument is given for the converse or for the uniqueness of continuous solutions of (2.3). This gap is load-bearing because the deep learning loss (3.1) minimizes the residual of (2.3); in the absence of uniqueness, a minimizer could converge to a spurious solution that is not the viscosity solution. The authors acknowledge in Section 5 that rigorous analysis of the implicit formula is future work, which is inconsistent with the theorems as stated.
[§2.1, Example 2.1] For H(p) = p^2, substituting into (2.3) gives u = -t(u_x)^2 + t u_x (2u_x) = t(u_x)^2, not u = t u_x as written in (2.15). The ODE u = t u_x has solutions u = C e^{x/t}, but the correct equation u = t(u_x)^2 admits a different family, e.g., u(x,t) = (x - 2Ct)^2/(4t) for t>0. The uniqueness argument in the example therefore applies to a different equation and does not establish uniqueness for the stated problem, even in the simplest quadratic case.
[§2.1, theorem statements] The phrase 'is the viscosity solution of (2.1) a.e.' is undefined: viscosity solutions are defined pointwise through test-function inequalities, not a.e. In addition, a merely continuous function need not have a gradient a.e. (e.g., continuous nowhere-differentiable functions), while (2.3) requires ∇u. The theorems should either assume u is locally Lipschitz (so ∇u exists a.e.) or otherwise specify the regularity needed for the formula and the conclusion.
[§2.1, proof of Theorem 2.1] In the proof of Theorem 2.1, equation (2.9) asserts that the Euler-Lagrange minimizer satisfies y* = x - t∇H(∇u), but this identity is not derived from the preceding variational problem without already using z* = ∇u, which is only established in (2.10). The proof has an order-of-dependencies issue; a rigorous derivation from the Hopf-Lax formula, including the required regularity of u, should be provided.

minor comments (6)

[§1 and §4] The title/abstract contain typos: 'EFFICIENTL Y', 'SOL VING', 'HAMIL TON-JACOBI'; in §4, 'for for N = 200,000 epochs' should be 'for N = 200,000 epochs'.
[§2.1, after (2.7)] The text refers to 'the Hopf-Lax formula (2.1)', but the Hopf-Lax formula is (2.4); the reference should be corrected.
[§2.2] The phrase 'the PMP formula (3.5)' in the discussion following (2.25) should refer to the PMP system (2.20), not (3.5).
[§3.3] The sentence 'The system of characteristic ODEs of (3.6) is given by' should refer to equation (3.4), not (3.6).
[§4.2, Example 4.9] The sentence 'where we can observe can observe that' contains a duplicated phrase.
[§3.2, Remark 3.1] The phrase 'the the implicit solution formula' contains a duplicated article.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the implicit formula is derived from characteristics and benchmarked against Hopf-Lax/Hopf; the main gaps are a missing converse/uniqueness proof and an erroneous example, not circular reductions.

full rationale

The paper's central derivation is self-contained rather than circular. Formula (2.3) is obtained directly from Hamilton's characteristic ODEs (2.2) by setting p = ∇u at the current point; it does not assume the Hopf-Lax or Hopf formulas. Theorems 2.1 and 2.2 then use the classical Hopf-Lax and Hopf representations as external benchmarks to verify that the viscosity solution satisfies the implicit formula. The deep learning loss (3.1) is simply the squared residual of the derived implicit equation, with no fitted physical constants and no supervised data, so the numerical solutions are not a renamed fit. The state-dependent Hamiltonian algorithm is an explicit piecewise-linear approximation, not a circular step. The paper's real weaknesses are rigor gaps: Theorems 2.1 and 2.2 prove only that the viscosity solution satisfies (2.3), and they do not prove uniqueness or the converse claim that any continuous solution of (2.3) is the viscosity solution; Section 5 explicitly defers rigorous analysis to future work. Moreover, Example 2.1 writes (2.15) as u = t u_x for H(p)=p^2, but substitution into (2.3) gives u = t u_x^2, so the subsequent ODE uniqueness argument applies to a different equation. These are correctness and proof-completeness concerns, not circularity, because no load-bearing step reduces to its own inputs by construction or by self-citation.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central formula is derived from the characteristic ODEs, which are standard. The theorems import the Hopf-Lax and Hopf representation formulas as black boxes. The main unproved input is the uniqueness of continuous solutions of the implicit equation. All other inputs are hand-chosen numerical hyperparameters.

free parameters (8)

Boundary regularization weight lambda = 0.1
Chosen in Section 4 to weight the boundary loss; not tuned per problem.
Softplus steepness beta = 100
Activation sigma(x) = (1/beta) log(1 + exp(beta x)); hand-picked to approximate ReLU while remaining smooth.
Network depth L = 5
Fixed across all experiments.
Network width = 64
Fixed across all experiments.
Collocation points per epoch M = 5000
Monte Carlo sample size for the loss integral.
Training epochs N = 200,000
Number of gradient steps for every experiment, reported in Section 4.
Initial learning rate and schedule = 1e-3, decay 0.99 on loss decrease
Hand-chosen schedule reported in Section 4.
Time step Delta t for state-dependent H = 0.1, 0.25, 0.5 in tests
Controls the piecewise linear characteristic approximation; first-order accuracy in Delta t is demonstrated.

assumptions (5)

domain assumption Hopf-Lax formula gives the viscosity solution for convex H and lower semicontinuous g.
Used in Theorem 2.1 proof; cited from [33, 5, 50].
domain assumption Hopf formula gives the viscosity solution for convex g and continuous H.
Used in Theorem 2.2 proof; cited from [33, 5].
domain assumption Characteristic ODEs (2.2) are valid and p = nabla u is constant along characteristics; the formula holds a.e. for viscosity solutions.
Standard theory from viscosity solutions and the method of characteristics, invoked throughout Section 2.
domain assumption The neural network approximation error and the Monte Carlo loss (3.3) are small enough that the trained network satisfies the implicit formula to the reported accuracy.
Assumed in the numerical sections; no convergence guarantee for the optimization is proved.
ad hoc to paper For state-dependent H, p is approximately constant on each time interval Delta t.
This is the piecewise linear approximation in Section 3.3; it is a modeling assumption, not a theorem.

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Cite this review

Pith. "Pith review of Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations." pith.science (2026). https://pith.science/paper/42LGUJF6

@misc{pith2026250119351,
  author       = {Pith},
  title        = {Pith review of: Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42LGUJF6}},
  note         = {Machine review of arXiv:2501.19351}
}

read the original abstract

This paper presents an implicit solution formula for the Hamilton-Jacobi partial differential equation (HJ PDE). The formula is derived using the method of characteristics and is shown to coincide with the Hopf and Lax formulas in the case where either the Hamiltonian or the initial function is convex. It provides a simple and efficient numerical approach for computing the viscosity solution of HJ PDEs, bypassing the need for the Legendre transform of the Hamiltonian or the initial condition, and the explicit computation of individual characteristic trajectories. A deep learning-based methodology is proposed to learn this implicit solution formula, leveraging the mesh-free nature of deep learning to ensure scalability for high-dimensional problems. Building upon this framework, an algorithm is developed that approximates the characteristic curves piecewise linearly for state-dependent Hamiltonians. Extensive experimental results demonstrate that the proposed method delivers highly accurate solutions, even for nonconvex Hamiltonians, and exhibits remarkable scalability, achieving computational efficiency for problems up to 40 dimensions.

Discussion (0). Continue with ORCID to comment.

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