Physics-informed Temporal Difference Metric Learning for Robot Motion Planning

Ahmed H Qureshi; Ruiqi Ni; Zherong Pan

REVIEW 2 major objections 7 minor 1 cited by

Physics-informed Temporal Difference Metric Learning for Robot Motion Planning

T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding a temporal difference loss and a metric-space parameterization to neural Eikonal solvers yields self-supervised motion planners that succeed in complex and unseen environments, reporting up to 99.4 percent…

desk verdict The paper's obstacle-normal loss (Eq. 6) is undefined where it acts due to clipping, yet the rest of the work is solid enough to warrant a careful referee rather than a desk reject. read the letter →

arxiv 2505.05691 v1 pith:H5GG5NXF submitted 2025-05-09 cs.RO cs.LG

Ruiqi Ni , Zherong Pan , Ahmed H Qureshi This is my paper · ORCID

classification cs.ROcs.LG

keywords robotmotionplanningEikonalequationtemporaldifferencelearningmetricself-supervisedneuralfieldsgeodesicdistanceBellmanoptimality

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 argues that self-supervised neural motion planners trained by solving the Eikonal equation fail in complex and unseen environments because they enforce only the equation's pointwise gradient condition and not the two further identities that a true solution carries: being an optimal value function and being a geodesic distance. The proposed remedy adds a temporal difference loss that enforces Bellman's principle over a finite time step, an obstacle-normal alignment loss, a causality weighting, and a metric-learning architecture that stores the travel time as a learned latent-space distance with an $\ell^1$/ $\ell^\infty$ combination. With these modifications the paper reports a 99.4 percent success rate in cluttered 3D scenes and 91 percent for a 12-DOF dual-arm robot in a confined cabinet, with planning times often below a tenth of a second. For a general reader the upshot is that a collision-free path can be drawn from a learned cost field by sampling, without expert demonstrations and without gradient descent at runtime.

What carries the argument

The central object is the learned travel-time field $T(q_s,q_g)$ whose gradient norm is constrained to equal the reciprocal ground-truth speed $1/S^\star(q)$, making it simultaneously an optimal value function and a geodesic distance. The mechanism is the composite loss $\mathcal{L}=(\lambda_E\mathcal{L}_E+\lambda_{TD}\mathcal{L}_{TD}+\lambda_N\mathcal{L}_N)\mathcal{L}_C$, where $\mathcal{L}_E$ is the Eikonal speed loss, $\mathcal{L}_{TD}$ enforces $T(q_s,q_g)=\Delta t/S^\star(q_g)+T(q_s,q_g+u_g^\star\Delta t)$ along the optimal direction, $\mathcal{L}_N$ aligns the field's gradient with obstacle normals near obstacles, and $\mathcal{L}_C=\exp(-\lambda_C T(q_s,q_g))$ enforces causality by prioritizing small values first. The metric space is parameterized as $T(q_s,q_g)=D(f_\theta(q_s),f_\theta(q_g))$ with $D(x,y)=\sum_i \max_j |x_{i,j}-y_{i,j}|$, an $\ell^1$-sum of $\ell^\infty$ terms chosen so multiple shortest paths are not collapsed into one chord.

What would settle it

Run the method on a maze with a deliberately narrow corridor, compute $d_{\mathrm{obs}}$ with a signed-distance routine that is known to be non-smooth along the medial axis, and compare the learned travel-time contours against a dense Fast Marching reference; if the contour error concentrates exactly where the finite-difference gradient of $S^\star$ disagrees with the analytic gradient used in the obstacle-normal loss, the differentiability assumption is the weak point. Alternatively, toggle the obstacle-normal loss in that corridor: failure to converge would indicate the term is receiving wrong normal directions.

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Extended reading notes

Core claim

The paper's central claim is that the Eikonal equation's solution should be read simultaneously as the optimal value function of an optimal control problem and as the geodesic distance of a Riemannian manifold whose metric is $1/S^\star(q)$, and that a neural travel-time field trained only with the pointwise Eikonal loss can satisfy the PDE at sampled points yet drift arbitrarily between them. The paper asserts that adding a temporal difference loss derived from a Taylor expansion along the optimal policy fixes the inter-sample drift, an obstacle-normal alignment term anchors early training near obstacles, a causality weight enforces one-way value propagation, and a metric-space parameterization $T(q_s,q_g)=D(f_\theta(q_s),f_\theta(q_g))$ with $D$ a sum of $\ell^\infty$ terms preserves triangle inequality, symmetry, and non-negativity while allowing multiple shortest paths. With these ingredients and environment conditioning through attention on the obstacle point cloud, the paper reports that the learned field solves the Eikonal equation accurately enough to support sampling-based MPC inference and to generalize to unseen environments across 2 to 12 degrees of freedom.

Load-bearing premise

The training pipeline assumes that the minimum workspace distance between the robot's geometry at configuration $q$ and the obstacles, $d_{\mathrm{obs}}(q,\mathcal{X}_{\mathrm{obs}})$, is differentiable in $q$ through differentiable forward kinematics; if that distance is kinked or approximated in narrow passages or high-DOF arms, the Eikonal, temporal difference, and obstacle-normal losses carry undefined or biased gradients.

Editorial extensions

If this is right

A travel-time field that is both a metric and an Eikonal solution lets the planner extract paths with sampling-based MPC alone, so inference needs no gradient computation and can recover from local inaccuracies through stochastic exploration.
Combining the Eikonal loss with the finite-step temporal difference loss is what suppresses spurious local minima: in the 2D maze ablation the full loss reaches error 0.08 versus 1.13 with only the Eikonal loss and 0.21 without the TD loss.
Environment conditioning through point-cloud attention makes the learned Eikonal solver generalizable: on unseen C3D scenes success is 99.2 percent and on unseen 7-DOF manipulator scenes 84.0 percent, close to seen-scene performance.
The same learned cost-to-go can serve other downstream planners such as cost-aware T-RRT-style sampling, because the field is a valid metric and not merely a local gradient.
In the 12-DOF real-world cabinet task the method reports 91 percent success in about 0.09 seconds on average, where prior self-supervised planners did not converge.

Reading between the lines

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

If the differentiability assumption on $d_{\mathrm{obs}}$ breaks at medial-axis kinks, the obstacle-normal loss in Eq. 6 should show exactly where: train the same architecture in a corridor whose width approaches the TD step and the finite-difference surrogate of $S^\star$; expect the contour error to spike where the two gradients disagree.
The paper's metric argument suggests the $\ell^1$/ $\ell^\infty$ combination is load-bearing, so a natural test is to replace only the distance $D$ with, say, a learned quasimetric while keeping every other loss term; if success falls on multi-connected workspaces, the multipath-preserving property is confirmed.
Because the paper reports weaker generalization to unseen Gibson homes, its own numbers imply the bottleneck is the environment encoder rather than the Eikonal loss; swapping the point-cloud encoder for another shape-conditioning architecture and re-measuring unseen-scene success would isolate that claim.
The TD step $\Delta t$ is tied to environment clutter (0.02 in open scenes, 0.005 for manipulators), which predicts a direct trade-off: any cluttered environment whose narrowest passage gives a time step below what the model can represent should force either more training epochs or a smaller $\Delta t$.

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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

Here's the takeaway: the paper's obstacle-normal alignment loss (Eq. 6) is undefined in the region where it matters, because of the clipping in Eq. 2. The stress-test note is correct. That's a real flaw, not a quibble. But the rest of the paper is better than the flaw, and the empirical story deserves scrutiny rather than a desk reject.

What's new: the authors frame the Eikonal solution as an optimal value function and a geodesic, then add a temporal difference loss to the usual Eikonal loss, plus a metric-learning architecture (blockwise L1/L∞) that respects triangle inequality and symmetry. They also condition on the environment point cloud with attention and use sampling-based MPC at inference. Each piece is motivated and ablated. The maze ablation shows every loss component helps, and the metric comparison against IQE, PQE, MRN, DN is fair. The main results are strong: 99.4% success on C3D, 91% on a 12-DOF dual-arm, with about 5-10x speedup over classical baselines.

The soft spots, in proportion. First, Eq. 6 is broken as written. S* is clipped, so inside the near-obstacle shell dobs

Second, the gradient-inference variant Ours-G only appears in the Gibson table. In the C3D and manipulator tables, we can't tell whether the gains come from the learned value function or from the MPC policy. The authors had the right idea in Gibson; they should report it everywhere.

Third, the baselines. NTF and P-NTF are upgraded with the same attention encoder as 'ours', which is reasonable for generalization testing, but it means the comparison isn't against the published versions. And the code isn't out yet, despite the abstract's repo link.

None of these are fatal to the central idea. The TD-Eikonal combination is sensible, the metric architecture is a legitimate contribution, and the authors' own conclusion is honest about generalization limits. The paper is useful to people working on self-supervised planning or learned cost functions. I'd give it a serious referee: ask for a corrected Eq. 6, Ours-G results everywhere, and code release.

My verdict: engage with the paper, but don't take its loss function at face value.

Referee Report

2 major / 7 minor

Summary. The paper presents a self-supervised method for robot motion planning that learns a travel-time function T(qs,qg) by solving the Eikonal equation. The training objective in Eq. (8) combines the standard Eikonal loss, a temporal difference loss enforcing Bellman optimality over a finite step, an obstacle-normal alignment loss, and a causality weight. The travel-time function is parameterized as a metric distance in a learned latent space using a blockwise L1/L-infinity norm, and the encoder is conditioned on environment point clouds through attention, which enables generalization to unseen environments. Planning at test time is performed with sampling-based MPC. Experiments on 2D mazes, Gibson indoor scenes, C3D cluttered 3D scenes, 7-DOF Franka manipulation, and a 12-DOF dual-arm real-world cabinet show improved success rates and lower planning times relative to prior self-supervised planners.

Significance. The paper makes a solid empirical case that adding a finite-horizon Bellman consistency term to the Eikonal loss improves the quality of the learned travel-time field, and that a metric-learning architecture with a non-Euclidean latent distance is beneficial. The ablation study in Table 2 quantifies the contribution of each loss component and of the metric choice, and the reported success rates on multi-DOF tasks are strong. The environment-conditioned attention mechanism is a useful step toward generalization to unseen scenes, and the promise of code release is a positive reproducibility signal. However, the current manuscript contains an undefined loss term in a load-bearing component, and the empirical gains are not fully disentangled from the MPC inference scheme, so the central claim of a more accurate Eikonal solution is not yet fully supported.

major comments (2)

[§4.1.2, Eq. (6)] The obstacle-normal alignment loss L_N is ill-defined on a set of positive measure. Since S* is defined by the clipping in Eq. (2), for any configuration with d_obs(q) < d_min, S* is constant at d_min/d_max and hence ∇S* = 0, while the weight (1−S*) is positive. The expression ∇S*/||∇S*|| is therefore 0/0 precisely in the near-obstacle region where the loss is intended to be active. The differentiability assumption on d_obs in §3.2 does not resolve this issue because clipping eliminates the gradient. The manuscript does not specify a surrogate, such as using the gradient of the unclipped distance or clamping the denominator, so the actual trained objective is underspecified. Since the ablation in Table 2 shows that removing L_N raises the maze error from 0.08 to 0.13, this is a load-bearing reproducibility and correctness concern.
[§5.2, Tables 1(b) and 1(c)] The Ours-G variant, which uses gradient-based path inference instead of MPC, is reported only for the Gibson environment in Table 1(a). In the C3D and 7-DOF manipulator experiments, the comparison of Ours (MPC) against NTF and P-NTF (gradient-based inference) is confounded by the inference mechanism. It is therefore possible that the higher success rates in Tables 1(b) and 1(c) come from MPC's stochastic exploration rather than from a more accurate learned value function. The authors should report Ours-G results for these tasks or explicitly disentangle the contribution of MPC from that of the learned Eikonal solution to support the central claim that the proposed losses and architecture improve the value function.

minor comments (7)

[§4.1.3, Eqs. (7)-(8)] The causality weight L_C = exp(-λ_C T(qs,qg)) multiplies all losses, including the Eikonal loss. While this is a known heuristic from Wang et al. (2024b), the paper does not analyze whether this output-dependent weighting changes the set of stationary points of the combined loss. Even though the true solution is still a stationary point, spurious minima may be introduced; a brief empirical or theoretical note would strengthen the claim that causality preservation is one of the three key enhancements.
[§4.2.1] The claim that the L1/L∞ metric 'preserves the geodesic structure' and supports multiple shortest paths is supported only by the illustration in Fig. 2. Please clarify whether this is a rigorous property of the proposed construction or an empirical observation, and provide a mathematical statement if available.
[§3.2 and §4.2.1] The hyperparameters d_min, d_max, a, and b in Eqs. (2) and (10) are never specified. Since these define the ground-truth speed and the latent space geometry, please include their values in Appendix C or in the code release to make the experiments reproducible.
[Appendix C.2] The hyperparameters λ_E, λ_TD, λ_N, λ_C, and Δt are given, but the dependence of Δt and λ_N on the environment is described only qualitatively; please provide the specific values used for each environment.
[§5.2 and figure captions] There are several typos and reference inconsistencies: 'Our-G' vs 'Ours-G' in §5.2; 'metirc' in §4.2.1; 'Demontration' in the Fig. 5 caption; and the P-NTF citation in §5 lists (Ni & Qureshi, 2023a) while the related work and §3.2 cite (Ni & Qureshi, 2023b).
[Fig. 2 caption] The caption begins with 'L1L2', which appears to be a leftover label; the figure should be clarified with respect to the L1 and L∞ components of the proposed metric.
[§4.3] The sampling-based MPC is described qualitatively, but the number of samples, horizon, and softmax temperature are not reported; please include these parameters in Appendix C.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central losses are anchored to an external distance-derived speed field, and headline accuracy is checked against external FMM ground truth.

full rationale

The derivation chain is self-contained rather than circular. In Eq. 2, the ground-truth speed S* is computed from external geometric obstacle distances, so the Eikonal loss (Eq. 3) and the TD loss (Eq. 5) are anchored to geometric input, not to the network's own predictions. The TD loss is a bootstrapped Bellman residual, which is a standard training objective rather than a hidden fit. The metric function D (Eq. 10) is imposed as an architectural constraint, not claimed as an empirical prediction; the symmetry and triangle-inequality properties follow directly from the definition of D, which is a design property rather than a derived result. The maze experiments are evaluated against external FMM ground truth (Table 2), and the C3D and manipulator success rates are not fitted parameters. Self-citations to NTFields and P-NTFields are used as prior basis and as baselines, not as an unverified uniqueness theorem or as a parameter fit. The paper does contain a genuine mathematical concern in Eq. 6, where after Eq. 2's clipping the term ∇S*/||∇S*|| is 0/0 in the near-obstacle region, and the authors themselves admit in Sec. 6 that the method 'struggles to generalize effectively to multiple, unseen environments in complex Gibson datasets'; however, these are correctness and reproducibility issues, not circular derivation, so they do not raise the circularity score.

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

The method does not introduce new physical entities or forces. Its contributions are a loss combination, a metric architecture, and an inference procedure. The load-bearing free parameters are the distance thresholds dmin and dmax, the tuned loss weights and TD step, and the unreported architecture and MPC hyperparameters. The key domain assumptions are differentiability of the signed distance field and the sufficiency of point-cloud conditioning for generalization.

free parameters (5)

dmin and dmax thresholds in Eq. 2 = not reported
These thresholds define the ground truth speed field S* and therefore affect every loss term and the value function. They are chosen by hand and their values are omitted from the paper.
Loss weights lambda_E, lambda_TD, lambda_N, lambda_C = 1e-2, 1e-3, 1e-3 for 3D; 1e-2, 1e-3, 2e-4 for manipulator; lambda_C = 0.5
The authors state the weights were chosen with cross-validation and tuned per environment type, making them fit to validation performance.
TD step Delta_t = 0.02 for 3D, 0.005 for manipulator
The step size is adjusted based on clutter and narrow passages, and the authors note that too large or too small values hurt training.
Latent dimensions a and b in Eq. 10 = not reported
The proposed metric D depends on the latent shape a times b, but the paper does not specify these dimensions or the encoder output size.
MPC inference hyperparameters = not reported
Action noise, horizon length, softmax temperature, number of rollouts, and time limits are not specified, though they directly affect success rate and planning time.

assumptions (5)

standard math The solution to the Eikonal equation equals the value function of the optimal control problem in Eq. 4 with dynamics qdot = u, ||u|| = 1, and cost integral ||qdot|| / S* dt.
This is a standard Hamilton-Jacobi equivalence. The paper provides an infinitesimal perturbation argument in Appendix A and uses it to justify both LE and LTD.
domain assumption dobs(q, Xobs), the minimum workspace distance between the robot and obstacles, is differentiable in q through differentiable forward kinematics.
Explicitly assumed in Sec. 3.2 and used in LE, LTD, and especially LN. Signed distance gradients can be non-smooth at boundaries and in narrow passages.
domain assumption A point cloud Xobs together with PointNext and attention conditioning provides enough information to approximate Eikonal solutions for unseen environments.
Sec. 4.2.2. The generalization claims depend on this representation capturing environment geometry sufficiently well.
ad hoc to paper The blockwise L1 over L-infinity metric in Eq. 10, composed with a learned encoder, yields a travel-time function that preserves the multiple-shortest-path geodesic structure of the Eikonal solution.
Motivated by Fig. 2 but not proven. The claim that this metric better captures Eikonal properties rests on this design choice and the maze experiments.
ad hoc to paper Reweighting the total loss by the causality weight exp(-lambda_C T) does not change the fixed point of the Eikonal solution while improving convergence.
Adapted from Wang et al. 2024b. The paper provides no analysis of how this reweighting affects the optimum, only empirical evidence that it helps training.

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Cite this review

Pith. "Pith review of Physics-informed Temporal Difference Metric Learning for Robot Motion Planning." pith.science (2026). https://pith.science/paper/H5GG5NXF

@misc{pith2026250505691,
  author       = {Pith},
  title        = {Pith review of: Physics-informed Temporal Difference Metric Learning for Robot Motion Planning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5GG5NXF}},
  note         = {Machine review of arXiv:2505.05691}
}

read the original abstract

The motion planning problem involves finding a collision-free path from a robot's starting to its target configuration. Recently, self-supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural networks and lead to efficient solutions. However, these methods struggle in complex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose a novel self-supervised temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning to preserve the Eikonal equation's essential geodesic properties. We demonstrate that our approach significantly outperforms existing self-supervised learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from 2 to 12 degrees of freedom (DOF).

Figures

Figures reproduced from arXiv: 2505.05691 by the authors.

**Figure 1.** The plots depict the solution to the Eikonal equation |∇qT(q)| = 1, T(0) = 0 with sampled green points as training data. In the top plot, the desired solution T(q) = |q| is shown. In contrast, the bottom plot demonstrates that the green points also satisfy |∇qT(q)| = 1, but without proper constraints, the solution deviates. The TD loss LTD ensures correctness by enforcing, for example, that for q1 > 0, T(q1) − T(q1… view at source ↗

**Figure 2.** The figure shows the geodesics of points A and B on a circle under different embeddings. With L2, the distance collapses into a line, causing overlap and ambiguity. In contrast, L1 transforms it into a diamond, preserving the geodesic structure and resolving ambiguity. with D(·, ·) being some metric function satisfying the three aforementioned properties. The definition of metric function D is crucial to the pe… view at source ↗

**Figure 3.** First row: We compare our method with FMM, NTF, and P-NTF. Middle row: We ablate our method by removing −LE, −LTD, −LN , and −LC . Third row: We replace our distance metric D with IQE, PQE, MRN, and DN. The first row in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Depiction of our (a) Gibson, (b) Cluttered 3D (C3D), and (c) 7-DOF Manipulator environments. We also illustrate multiple trajectories planned by our method between different start and goal pairs. It can be seen that our method finds smooth trajectories while avoiding c…

**Figure 5.** Figure 5: Demontration of a path planned by our method to navigate real-world cabinet environment using a 12-DOF dual-arm robot. This particular trajectory was planned in 0.11 seconds. and unseen environments, while Appendix. B details performances on seen and unseen tasks sepa…

**Figure 6.** Figure 6: Additonal two mazes results. First row: We compare our method with FMM, NTF, and P-NTF. Middle row: We ablate our method by removing −LE, −LTD, −LN , and −LC . Third row: We replace our distance metric D with IQE, PQE, MRN, and DN. In [PITH_FULL_IMAGE:figures/full_fig…

**Figure 7.** Figure 7: Visualization of Gibson environments, including multiple trajectories planned by our method between various start and goal pairs. It can be observed that our method generates smooth, collision-free trajectories, effectively navigating through the environments while a…

**Figure 8.** Figure 8: Another path planned by our method to navigate real-world cabinet environment using a 12-DOF dual-arm robot. This trajectory is inferred in 0.1 seconds. C.1 TRAINING DETAILS This section summarizes our training procedure. Our training data comprises randomly sampled ro…

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@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should ...
[63]

\@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...
[64]

@open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

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work page 2023