{"id":"92e2a6d3-7f5a-46c8-b552-cf014d752bde","arxiv_id":"2608.11480","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A simple forward-trajectory steering scheme for collocation sampling lets PINNs solve high-dimensional Hamilton-Jacobi reachability problems with accuracy comparable to or better than MPC-supervised solvers.","lead":"This paper introduces STEER2REACH, a physics-informed neural network solver for Hamilton-Jacobi reachability that samples training points along forward trajectories steered by the current estimate of the safety value function. It reports competitive safety performance and lower value-function error than an MPC-guided state-of-the-art solver on most benchmarks, with one clear failure case.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Safety metrics are evaluated against a disturbance chosen by the learned value function itself, so the advertised safety comparison is not yet validated against a true worst-case adversary.","rationale":"The reader's weakest assumption focuses on rollout fidelity of the forward SDE, which is indeed a real and directly evidenced limitation: F1Tenth fails precisely because rollouts deviate from the true system. I do not dispute that. However, I elevate a second, less visible premise that appears in Sec. V-A: the evaluation labels h*_theta use the learned disturbance d_theta as the adversary. This premise enters every safety metric in Table I, including the successful benchmarks, and it is not examined anywhere in the paper. Because the central claim is specifically about safety performance, failure of this premise would undermine the comparison even where rollout fidelity is adequate. The F1Tenth failure also makes the danger concrete: if V_theta can be inaccurate enough to bias sampling, it can also be inaccurate enough to select a non-adversarial d_theta. The paper's overall conclusion may survive a corrected evaluation, so the reader's CONDITIONAL verdict remains appropriate, but the conditions should include independent validation of the adversarial disturbance, not only correction of the abstract's scope.","tokens_in":17845,"tokens_out":11012,"duration_ms":107993,"concrete_test":"For vertical drone, pursuit-evade, and publisher-subscriber, recompute the safety evaluation (precision, IoU, and TV) for all four methods using the true worst-case disturbance d* obtained from the ground-truth value function, instead of the learned d_theta, while keeping the learned controller u_theta unchanged. If any reported TV or IoU shifts by more than 1 percentage point, the Table I safety comparison is an artifact of the self-consistent adversarial evaluation; if all metrics are stable, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The safety metrics in Table I are computed with respect to h*_theta(x) = 1{J(x,0,u_theta,d_theta)>0} as defined in Sec. V-A. Here d_theta is the argmin in Eq. (13) computed from the learned value function V_theta, not the true worst-case disturbance d* defined by the true value function in Eq. (8). Because D is a box, d_theta is bang-bang in the sign of <grad V_theta, f>, so a modest error in V_theta can flip the chosen disturbance from adversarial to non-adversarial. Precision, IoU, and TV are then measured against a safety label generated by the very network under evaluation. The paper calls d_theta an approximation but provides no bound, no sensitivity analysis, and no comparison with d* on the three benchmarks where ground truth is available. This is load-bearing for the central claim because the headline contribution is competitive or improved safety metrics. The F1Tenth discussion in Sec. V-D shows that when V_theta is inaccurate, trajectory-based sampling degrades; the same inaccuracy also corrupts the adversary used in the evaluation, so even the successful benchmarks currently rest on an unvalidated evaluation premise.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"STEER2REACH (S2R) proposes a lightweight adaptive collocation-sampling scheme for physics-informed neural network (PINN) solvers of Hamilton-Jacobi reachability problems. In place of uniform, residual-based, or MPC-guided sampling, S2R generates collocation points by Euler-Maruyama rollouts of a forward SDE in which the control and disturbance are the argmax/argmin of the Hamiltonian computed from the current learned value function, plus injected noise. The training loop is formulated as repeated gradient descent on the PDE-residual loss under the induced sampling measure. The paper reports experiments on five benchmarks (2D vertical drone, 3D pursuit-evade, 7D F1Tenth, 13D quadrotor, and 40D publisher-subscriber), comparing S2R with vanilla PINNs, RAD-PINNs, and MPC-guided DeepReach. The central claim is that S2R matches or improves upon the state-of-the-art MPC-DeepReach on safety metrics while achieving lower relative L2 error on problems with ground truth, without multi-stage training or MPC-based supervision. The paper also contains a candid discussion of a substantial performance gap on F1Tenth and of the sensitivity of trajectory-rollout sampling to rollout accuracy.","tokens_in":18016,"tokens_out":10033,"duration_ms":96259,"significance":"If the empirical claims are valid, S2R is a practically meaningful simplification: it removes the need for MPC regression data, curriculum learning, and multi-stage fine-tuning while achieving relative L2 errors that are 1.3x to 5.7x lower than MPC-DeepReach on the three benchmarks with ground truth, and comparable or better safety metrics on four of the five problems. The paper deserves credit for releasing code, for running ablations on the hard constraint parameterization and on the two new hyperparameters sigma and dt, for a scalability study up to 500 dimensions, and for explicitly discussing the F1Tenth failure rather than omitting it. The main unresolved concern is that the safety evaluation is not independent of the learned value function: the 'true' safety labels are generated using the disturbance that is optimal for the learned value, not for the true value. The RL2 comparisons, which are independent of that issue, provide the strongest evidence for the usefulness of the method.","major_comments":[{"comment":"The safety ground-truth label is computed as h*_theta(x) = 1{J(x,0,u_theta,d_theta)>0}, where d_theta is the argmin in Eq. (13) obtained from the learned value function V_theta. Since D is a box, d_theta is bang-bang in the sign of the gradient inner product, so a modest error in V_theta can switch the chosen disturbance from adversarial to non-adversarial and thereby inflate the reported precision, IoU, and TV. The paper calls d_theta an approximation but gives no error bound, no sensitivity analysis, and no comparison with the true worst-case disturbance d* on the three benchmarks where ground-truth V is available (vertical drone, pursuit-evade, publisher-subscriber). This issue is load-bearing because the central claim of 'competitive or improved safety metrics' rests on these labels. I request a re-evaluation of the safety metrics using a disturbance that is independent of the learned network (e.g., the true d* from ground-truth V, or an independent adversarial-search procedure), or, at minimum, a quantitative sensitivity study showing that the reported metrics are stable under perturbations of V_theta.","section":"Sec. V-A, Eq. (13)"},{"comment":"The metric called 'Precision' is defined as TP/(TP+FN), which is recall, not precision. Standard precision is TP/(TP+FP). As defined, this metric does not penalize false positives and can therefore make an over-optimistic safety classifier look better than it is. Since precision is one of the headline safety metrics in Table I, the formula should be corrected (or the metric renamed), and ideally the true precision should also be reported. This is a factual error in the evaluation protocol, and it directly affects the interpretation of the numerical comparisons.","section":"Sec. V-A, metric definitions"},{"comment":"The abstract states that S2R 'achieves competitive--and in some cases improved--performance on safety metrics across a range of reachability benchmarks.' This is contradicted by the F1Tenth row of Table I, where S2R reports precision 0.7500 vs 0.9794 and IoU 0.7364 vs 0.9603 against MPC-DeepReach, and by the authors' own Sec. V-D discussion. The paper is transparent about this in the body, but the abstract and the conclusion should be amended to scope the claim, e.g., by explicitly excluding F1Tenth or by saying that performance is competitive on four of the five benchmarks. The F1Tenth discussion also states that 'any inaccuracies in the rollouts directly affect where the PDE residual is evaluated,' which is a fundamental limitation of the trajectory-steered sampling premise that should be reflected in the paper's claims.","section":"Abstract, Sec. V-C, Sec. V-D"}],"minor_comments":[{"comment":"The line 'Define L_S2R,j(theta) = L_VI(theta_j) + lambda L_bnd(theta_j)' is notationally inconsistent because the right-hand side does not depend on the argument theta. It should read L_VI(theta) + lambda L_bnd(theta) evaluated at theta = theta_j, or the theta argument should be removed.","section":"Algorithm 1, line 21"},{"comment":"The publisher-subscriber problem is described as defining a 'target set' in the final paragraph of Appendix B, while the paper elsewhere refers to L as the 'failure set.' Since the paper's parameterization and value-function inequality V(x,t) <= l(x) are specific to the avoid formulation, please clarify whether publisher-subscriber is treated as a reach or avoid problem and reconcile the terminology.","section":"Appendix B, publisher-subscriber"},{"comment":"The implementation paragraph says the methods are 'effectively under-sampled compared to MPC-DeepReach' because MPC-DeepReach also trains on regression data. This is a reasonable point, but the phrase 'We match the total number of collocation points' is confusing when immediately followed by a statement that the methods are under-sampled; please clarify which quantity is matched and which is not.","section":"Sec. V-B"},{"comment":"The three post-processing strategies (clipping, bounding, filtering) are described qualitatively, but no quantitative results are shown. A short table or figure quantifying the resulting bias and the failure to recover performance would strengthen the discussion, especially because this section is the main evidence for the method's limitation on hybrid dynamics.","section":"Sec. V-D"},{"comment":"The runtime unit is given as 'Runtime[m]' but never defined; please state explicitly that m denotes minutes, and consider reporting on comparable hardware and with comparable total training examples.","section":"Table I caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for the journal and the core idea is promising, but the safety evaluation needs to be made independent of the learned adversary before the central claims can be accepted. The RL2 results are the most convincing part of the paper and should be emphasized. The F1Tenth discrepancy is honestly discussed and is not by itself a reason for rejection if the claims are scoped correctly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely simple idea that works on most benchmarks, but the abstract promises more than the data delivers, and there is one evaluation choice worth fixing before publication.\n\nWhat's new: S2R uses the current value function to pick control and disturbance, rolls out the forward SDE with those policies plus noise, and uses those trajectory states as the collocation distribution for the HJI-VI residual. That is a real and natural extension of the BSDE-sampling idea from their prior work, and it avoids the multi-stage training and MPC supervision that MPC-DeepReach needs. The paper does the comparison honestly: same architecture, matched collocation counts, three seeds, and it reports standard deviations. On vertical drone, pursuit-evade, quadrotor, and publisher-subscriber, S2R is competitive with MPC-DeepReach, and it beats it on RL2 by 1.3x–5.7x on the three problems with ground truth. The ablation on the constraint function and the noise/discretization sensitivity is useful.\n\nSoft spots. The F1Tenth result is substantially worse (precision 0.75 vs 0.98, IoU 0.74 vs 0.96), and the abstract's 'competitive, and in some cases improved' hides that. The paper does discuss F1Tenth at length, but the abstract should be qualified. More important: the safety metrics are computed against a ground-truth label that uses d_theta, the disturbance induced by the learned value function, not the true worst-case disturbance d*. That only bites on pursuit-evade, which is the one benchmark with a disturbance, but it is exactly the benchmark where the paper claims improved TV. The paper gives no bound or sensitivity analysis showing d_theta is close to d*. Even with low RL2, a small gradient error can flip a bang-bang disturbance. I'd ask for a comparison against d* on pursuit-evade, or at least a sensitivity analysis. The RL2 claim is also based on only three problems; the other two have no ground truth, which the paper notes but the abstract does not.\n\nVerdict: conditional to accept after the abstract is fixed and the pursuit-evade evaluation is checked. The method is worth a serious referee: it is simple, reproducible in principle, and opens a useful direction. I'd cite it if I worked on PINN reachability.","headline":"A simple and mostly effective adaptive sampling scheme for PINN reachability, undermined by an overclaiming abstract and a potentially circular safety metric on the one adversarial benchmark.","tokens_in":18640,"tokens_out":4682,"would_cite":true,"duration_ms":39301,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Steering forward trajectories under the current value function's optimal control produces safety metrics competitive with, and sometimes better than, an MPC-guided solver, while cutting relative L2 error by up to 5.7x.","keywords":["Hamilton-Jacobi reachability","physics-informed neural networks","adaptive collocation sampling","backward reachable tube","safety value function","forward trajectory steering","high-dimensional control","stochastic differential equations"],"falsifier":"Run S2R on a stiff or hybrid-dynamics control benchmark using an accurate implicit integrator for the rollouts instead of the Euler-Maruyama step with $N_{\\mathrm{traj}}=50$; if the gap to MPC-DeepReach on safety metrics and RL2 closes, rollout fidelity is the mechanism, and if the gap persists, the limit lies elsewhere in the sampling strategy.","tokens_in":17487,"feed_emoji":"🎯","tokens_out":19118,"duration_ms":134726,"temperature":0.7,"pith_summary":"Hamilton-Jacobi reachability is the standard way to certify that a control system will avoid unsafe states, but solving its high-dimensional PDE is slow, and recent physics-informed neural network (PINN) solvers have needed complicated training pipelines to work. This paper claims that the key difficulty is not the network but where the PDE residual is sampled, and that a cheap adaptive sampling rule is enough. The rule is to take the current learned value function, compute the control and disturbance it says are optimal, roll the system forward under those signals with a little noise, and train on the states those rollouts visit. The resulting method, called STEER2REACH (S2R), matches or improves the safety metrics of a state-of-the-art model predictive control (MPC)-guided solver on four of five benchmarks and reduces relative L2 error by 1.3x, 2x, and 5.7x on the three problems with known ground truth, all without multi-stage training or MPC supervision. If this is right, the expensive machinery of curriculum learning and model-predictive-control-generated labels is not necessary for accurate high-dimensional reachability value functions.","feed_headline":"Simple trajectory steering rivals costly reachability solvers","feed_subtitle":"It matches or beats a model-predictive-control solver on four of five benchmarks, with up to 5.7x lower error.","key_machinery":"The central object is the forward-SDE sampling measure $\\mu_\\theta = \\mathrm{Law}((X^{x;\\theta}_t, t))$, where $X^{x;\\theta}$ is the trajectory generated by Euler-Maruyama from the drift $f(x,u_\\theta,d_\\theta)$ plus additive noise $\\sigma\\,dW$, and $u_\\theta,d_\\theta$ are the control and disturbance induced by the current value function through the Hamiltonian argmax/argmin in equations (12)-(13). This measure replaces both the uniform collocation distribution of vanilla PINNs and the MPC-regression dataset of MPC-guided solvers. A second piece is the hard-constrained parameterization $V_\\theta(x,t)=\\ell(x)-(T-t)\\rho(\\phi_\\theta(x,t))$, which enforces the terminal condition and the inequality $V_\\theta\\le\\ell$ by construction and removes the boundary loss. The machinery works by continuously re-shaping the residual evaluation toward the part of state-time space the value function is actively controlling, and away from regions that do not matter.","core_discovery":"The central claim is that a simple forward-trajectory sampling scheme can replace the complex sampling strategies used by current PINN-based reachability solvers. S2R defines a sampling distribution from the law of a stochastic differential equation whose drift uses the control and worst-case disturbance that the current value function $V_\\theta$ would choose: $u_\\theta$ maximizes and $d_\\theta$ minimizes the inner product $\\langle \\nabla V_\\theta, f(x,u,d)\\rangle$. Euler-Maruyama integration of this SDE from uniformly sampled initial states, with Gaussian noise of strength $\\sigma$, produces collocation points that concentrate where the value function is actively steering; the PINN residual is then evaluated at those points. The paper demonstrates on 2D, 3D, 7D, 13D, and 40D benchmarks that this procedure yields safety-level-set metrics (precision, IoU, true volume) competitive with MPC-DeepReach, and better relative L2 error where ground truth is available; on the 40D publisher-subscriber problem it is the best method by all three safety metrics. The one clear miss is the 7D F1Tenth hybrid-dynamics problem, which the paper attributes to rollout inaccuracy biasing the collocation measure. The paper also shows that a hard constraint $V_\\theta(x,t) \\le \\ell(x)$, encoded through $V_\\theta = \\ell(x) - (T-t)\\rho(\\phi_\\theta)$, improves both PINN and S2R performance but is not the main source of S2R's advantage.","pith_inferences":["Not in the paper but directly suggested: any PDE solver whose accuracy hinges on collocation placement could adopt the same 'steer a surrogate process by the current solution, train where it goes' rule, so the idea is portable beyond HJ reachability.","The F1Tenth failure implies a testable boundary: the method's validity is tied to rollout fidelity, so stiff or hybrid systems that defeat the simple forward integrator are the natural first place to look for counterexamples.","Since the paper fixes the noise at $\\sigma=0.01$ and $N_{\\mathrm{traj}}=50$ across all benchmarks but its own ablation shows the best noise is problem-dependent, per-problem tuning is a plausible untested source of further gains.","The clipping/bounding/filtering results suggest a design principle for trajectory-based sampling measures: post-processing must preserve the measure induced by the true dynamics, a constraint that other adaptive sampling schemes would also need to respect."],"forward_implications":["On the 2D vertical drone, 3D pursuit-evade, 13D quadrotor, and 40D publisher-subscriber benchmarks, S2R matches or exceeds the safety-level-set metrics (precision, IoU, true volume) of MPC-DeepReach; on the 40D problem it is the top performer.","Relative L2 error against ground truth is lower for S2R than for MPC-DeepReach by 1.3x (pursuit-evade), 2x (drone), and 5.7x (publisher-subscriber).","S2R's training loop is ordinary Adam on the PINN residual with the steered sampling distribution; it uses no MPC regression loss, no curriculum, and no supervised pre-training, and it can drop the boundary loss entirely when the hard-constrained parameterization is used.","The 7D F1Tenth hybrid problem is the exception: S2R trails MPC-DeepReach, and the paper shows that clipping, bounding, or filtering rollout states to keep them in the domain does not fix the bias introduced by rollout error.","S2R scales better than uniform PINN sampling in the publisher-subscriber family, where its relative L2 advantage grows as the state dimension increases from 2 to 500."],"supporting_citations":[{"why":"Establishes the time-dependent Hamilton-Jacobi formulation of reachable sets that defines the value function and the HJI-VI being solved.","marker":"[1]"},{"why":"The MPC-guided DeepReach solver used as the main state-of-the-art baseline; supplies the benchmark suite, the comparison numbers, and the training codebase.","marker":"[7]"},{"why":"DeepReach, the curriculum-based PINN reachability solver whose loss and benchmarks frame the comparison.","marker":"[9]"},{"why":"Exact safety boundary condition parameterization that motivates S2R's hard constraint removing the boundary loss.","marker":"[10]"},{"why":"Prior work showing forward SDE trajectory steering can guide PINN sampling; S2R builds directly on this idea.","marker":"[16]"},{"why":"The residual-based adaptive sampling (RAD) method used as a baseline and as the main contrast to trajectory-based sampling.","marker":"[19]"},{"why":"Reflected BSDE theory connecting obstacle problems to forward-backward SDEs, the theoretical motivation for S2R's forward SDE.","marker":"[20]"},{"why":"Extends reflected BSDE theory to Isaacs stochastic differential games, justifying the control-versus-disturbance steering in S2R.","marker":"[21]"},{"why":"The released DeepReach codebase used as-is to train the MPC-DeepReach baseline for fair comparison.","marker":"[27]"}],"fun_headline_variants":["Steering forward trajectories sharpens reachability PINNs","Lightweight trajectory steering rivals heavy MPC solvers","S2R: simple sampling for high-dim reachability","Forward steering cuts reachability error up to 5.7x"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the simple forward-simulation rollouts, driven by the current value function's optimal control and worst-case disturbance, faithfully represent where the true system actually goes; if the rollouts are inaccurate, the PDE residual is evaluated in the wrong places and the training signal degrades, exactly as the paper reports for the F1Tenth benchmark.","fun_headline_variants_meta":{"raw":{"variants":["Steering forward trajectories sharpens reachability PINNs","Lightweight trajectory steering rivals heavy MPC solvers","S2R: simple sampling for high-dim reachability","Forward steering cuts reachability error up to 5.7x"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1699,"prompt_tokens":1091,"completion_tokens":608,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":542}},"tokens_in":707,"tokens_out":608,"duration_ms":5922,"temperature":1.0,"reasoning_tokens":542,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:12:18.924278+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run S2R on a stiff or hybrid-dynamics control benchmark using an accurate implicit integrator for the rollouts instead of the Euler-Maruyama step with $N_{\\mathrm{traj}}=50$; if the gap to MPC-DeepReach on safety metrics and RL2 closes, rollout fidelity is the mechanism, and if the gap persists, the limit lies elsewhere in the sampling strategy.","supporting_citations":[{"cited_title":"Integration matters for learning PDEs with backward SDEs,","cited_arxiv_id":null,"evidence_quote":"Prior work showing forward SDE trajectory steering can guide PINN sampling; S2R builds directly on this idea."},{"cited_title":"A time-dependent hamilton- jacobi formulation of reachable sets for continuous dynamic games,","cited_arxiv_id":null,"evidence_quote":"Establishes the time-dependent Hamilton-Jacobi formulation of reachable sets that defines the value function and the HJI-VI being solved."},{"cited_title":"DeepReach: A deep learning approach to high-dimensional reachability,","cited_arxiv_id":null,"evidence_quote":"DeepReach, the curriculum-based PINN reachability solver whose loss and benchmarks frame the comparison."},{"cited_title":"Exact imposition of safety boundary conditions in neural reachable tubes,","cited_arxiv_id":null,"evidence_quote":"Exact safety boundary condition parameterization that motivates S2R's hard constraint removing the boundary loss."},{"cited_title":"A comprehensive study of non-adaptive and residual-based adaptive sampling for physics- informed neural networks,","cited_arxiv_id":null,"evidence_quote":"The residual-based adaptive sampling (RAD) method used as a baseline and as the main contrast to trajectory-based sampling."},{"cited_title":"Reflected solutions of backward sde’s, and related obstacle problems for pde’s,","cited_arxiv_id":null,"evidence_quote":"Reflected BSDE theory connecting obstacle problems to forward-backward SDEs, the theoretical motivation for S2R's forward SDE."},{"cited_title":"Stochastic differential games with reflection and related obstacle problems for isaacs equations,","cited_arxiv_id":null,"evidence_quote":"Extends reflected BSDE theory to Isaacs stochastic differential games, justifying the control-versus-disturbance steering in S2R."},{"cited_title":"DeepReach Repository,","cited_arxiv_id":null,"evidence_quote":"The released DeepReach codebase used as-is to train the MPC-DeepReach baseline for fair comparison."}],"review_version":1}