{"id":"9274be8a-8b2e-47f3-acdd-52ae2344adc5","arxiv_id":"2501.19351","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"The paper learns approximate viscosity solutions of Hamilton-Jacobi equations by minimizing the residual of an implicit characteristic formula instead of the PDE residual.","lead":"This paper derives an implicit equation that Hamilton-Jacobi equation solutions should satisfy, and trains neural networks to solve that equation. The authors report accurate results on convex, nonconvex, and state-dependent test problems, including some in 40 dimensions, which could lead to practical solvers for high-dimensional control and motion-planning problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 2.1 and 2.2 prove only the forward direction, that the viscosity solution satisfies (2.3); they never prove uniqueness or the converse, and the Example 2.1 uniqueness argument is based on a mis-derived formula.","rationale":"The reader's weakest assumption is that (2.3) has a unique continuous solution; my reading of the proofs confirms the same gap. Theorems 2.1 and 2.2 are one-directional: they demonstrate that the classical viscosity solution satisfies the implicit formula, not that a solution of the implicit formula must be the viscosity solution. This is the load-bearing step for the deep learning objective, since minimizing the residual of (2.3) can only select the viscosity solution if the equation itself has no other solutions. The manuscript's own Section 5 states that rigorous analysis of the implicit solution formula remains future work, which is a direct admission that the converse/uniqueness question is unresolved. The algebraic error in Example 2.1 strengthens the concern: the paper's simplest illustration of uniqueness is computed incorrectly, so it cannot be cited as supporting evidence. I found no independent support that would offset this gap: there is no formal proof assistant verification, no reproducible code release, and no parameter-free derivation establishing uniqueness. The numerical results are suggestive but, as the reader notes, lack baselines and error bars, and they cannot settle a uniqueness question. I do not think the paper should be accepted in its current form, and my concern aligns with the reader's verdict rather than changing it.","tokens_in":20550,"tokens_out":11598,"duration_ms":119042,"concrete_test":"Run a 1D check with H(p)=p^2/2 and g(x)=x^2/2. Characterize all Lipschitz functions on R x [0,T] satisfying (2.3) a.e. with u(x,0)=g by solving the fixed-time ODE u=t(u')^2/2+(x-tu')^2/2, e.g. by factoring its x-derivative. Test whether the non-viscosity branch p=x/t+C can be patched to the viscosity branch p=x/(1+t) across a curve while preserving continuity in x and t and the initial condition. If a patched solution exists, Theorem 2.1 is false; if not, repeat with g(x)=|x| and on a bounded domain with periodic boundary conditions to probe whether the boundary condition restores the claimed uniqueness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the implicit equation (2.3) has a unique continuous solution over the whole spacetime domain and that this solution is the viscosity solution. The proofs of Theorems 2.1 and 2.2 start from the Hopf-Lax or Hopf representation and show that the viscosity solution satisfies (2.3). They do not show that an arbitrary continuous function satisfying (2.3) is the viscosity solution, nor that such a function is unique. Without uniqueness, the loss functional (3.1) may have zero objective at spurious solutions, so a neural minimizer is not guaranteed to recover the viscosity solution. The authors explicitly defer this analysis in Section 5, calling rigorous analysis of the implicit formula future work. A further concrete problem appears in Example 2.1: for H(p)=p^2, formula (2.3) gives u=t(u_x)^2, not u=t u_x as stated in (2.15). The subsequent ODE analysis and uniqueness argument therefore apply to a different equation, so the example does not even support the claimed uniqueness in the simplest case. The theorem statements also assert 'a.e.' without specifying the regularity of u needed to define ∇u, since a merely continuous function need not have a pointwise gradient.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20846,"tokens_out":10310,"duration_ms":89188,"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":[{"comment":"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.","section":"§2.1, Theorems 2.1 and 2.2"},{"comment":"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.","section":"§2.1, Example 2.1"},{"comment":"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.","section":"§2.1, theorem statements"},{"comment":"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.","section":"§2.1, proof of Theorem 2.1"}],"minor_comments":[{"comment":"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'.","section":"§1 and §4"},{"comment":"The text refers to 'the Hopf-Lax formula (2.1)', but the Hopf-Lax formula is (2.4); the reference should be corrected.","section":"§2.1, after (2.7)"},{"comment":"The phrase 'the PMP formula (3.5)' in the discussion following (2.25) should refer to the PMP system (2.20), not (3.5).","section":"§2.2"},{"comment":"The sentence 'The system of characteristic ODEs of (3.6) is given by' should refer to equation (3.4), not (3.6).","section":"§3.3"},{"comment":"The sentence 'where we can observe can observe that' contains a duplicated phrase.","section":"§4.2, Example 4.9"},{"comment":"The phrase 'the the implicit solution formula' contains a duplicated article.","section":"§3.2, Remark 3.1"}],"recommendation":"reject","confidential_remarks":"The central theoretical claim is not proven as stated, and the Example 2.1 error is a concrete flaw in an argument meant to establish uniqueness. The authors would need to either prove uniqueness/converse for (2.3) under the stated assumptions or substantially weaken the theoretical claims and reposition the paper as an empirical method with a necessary-condition formula. Given the paper's framing around a validated 'implicit solution formula', this gap is fundamental rather than a local fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nNow 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.\n\nThere 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.\n\nOn 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.\n\nBottom 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.","headline":"Clean idea and promising numerics, but the central uniqueness theorem is unproven and Example 2.1 contains a concrete formula error.","tokens_in":21359,"tokens_out":3517,"would_cite":false,"duration_ms":32713,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M25","68T07","35C99"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single implicit formula, learned by a neural network, solves high-dimensional Hamilton–Jacobi equations without grids or Legendre transforms.","keywords":["Hamilton-Jacobi equations","implicit solution formula","viscosity solutions","deep learning","method of characteristics","high-dimensional PDEs","curse of dimensionality","optimal control"],"falsifier":"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.","tokens_in":20341,"feed_emoji":"📐","tokens_out":4502,"duration_ms":41659,"temperature":0.7,"pith_summary":"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.","feed_headline":"Implicit formula solves high-dimensional Hamilton-Jacobi PDEs","feed_subtitle":"A neural network learns the formula without grids or Legendre transforms, reaching 40 dimensions at near-constant cost.","key_machinery":"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.","core_discovery":"The paper's central object is the identity\n$$u(x,t)=-tH(\\nabla u)+t\\nabla u^\\top\\nabla H(\\nabla u)+g(x-t\\nabla H(\\nabla u)),$$\nwhich 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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines viscosity solutions, the notion of solution the implicit formula is claimed to select.","marker":"[14]"},{"why":"Provides the Hopf formula that the implicit formula is shown to coincide with in the convex-initial-data case.","marker":"[33]"},{"why":"Establishes validity conditions for Hopf's formulas, used in Theorem 2.2.","marker":"[5]"},{"why":"Supplies the Hopf-Lax representation theory used in Theorem 2.1.","marker":"[50]"},{"why":"Earlier causality-free, curse-of-dimensionality-avoiding algorithms that the proposed method improves upon by removing Legendre transforms.","marker":"[20]"},{"why":"Provides the state-dependent and nonconvex Hamilton-Jacobi test problems used to validate the time-marching extension.","marker":"[13]"},{"why":"Classical high-order ENO schemes and benchmark examples, especially the nonconvex problems in Section 4.2.","marker":"[60]"},{"why":"The closely related prior neural implicit-form approach for conservation laws that this paper extends to viscosity solutions of HJ equations.","marker":"[75]"}],"fun_headline_variants":["Neural implicit formula cracks HJ PDEs in 40 dimensions","Skip the Legendre transform: neural nets solve HJ PDEs","Hamilton-Jacobi PDEs solved by a single learned identity","Deep learning replaces grids for high-dimensional HJ equations","Implicit solution formula: HJ PDEs without characteristics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Neural implicit formula cracks HJ PDEs in 40 dimensions","Skip the Legendre transform: neural nets solve HJ PDEs","Hamilton-Jacobi PDEs solved by a single learned identity","Deep learning replaces grids for high-dimensional HJ equations","Implicit solution formula: HJ PDEs without characteristics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1561,"prompt_tokens":957,"completion_tokens":604,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":524}},"tokens_in":573,"tokens_out":604,"duration_ms":5930,"temperature":1.0,"reasoning_tokens":524,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:27:00.332668+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines viscosity solutions, the notion of solution the implicit formula is claimed to select."},{"cited_title":"Hopf, Generalized solutions of non-linear equations of first order , Journal of Mathematics and Mechanics, 14 (1965), pp","cited_arxiv_id":null,"evidence_quote":"Provides the Hopf formula that the implicit formula is shown to coincide with in the convex-initial-data case."},{"cited_title":"Bardi and L","cited_arxiv_id":null,"evidence_quote":"Establishes validity conditions for Hopf's formulas, used in Theorem 2.2."},{"cited_title":"Lions, Generalized solutions of hamilton-jacobi equations , (No Title), (1982)","cited_arxiv_id":null,"evidence_quote":"Supplies the Hopf-Lax representation theory used in Theorem 2.1."},{"cited_title":"Darbon and S","cited_arxiv_id":null,"evidence_quote":"Earlier causality-free, curse-of-dimensionality-avoiding algorithms that the proposed method improves upon by removing Legendre transforms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the state-dependent and nonconvex Hamilton-Jacobi test problems used to validate the time-marching extension."},{"cited_title":"Osher and C.-W","cited_arxiv_id":null,"evidence_quote":"Classical high-order ENO schemes and benchmark examples, especially the nonconvex problems in Section 4.2."},{"cited_title":"Zhang, T","cited_arxiv_id":null,"evidence_quote":"The closely related prior neural implicit-form approach for conservation laws that this paper extends to viscosity solutions of HJ equations."}],"review_version":1}