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On the Numerical Reliability of Differentiable Physics-Based Optimization for Robotic Material Manipulation

Xintong Yang, Minglun Wei, Yu-Kun Lai, Ze Ji

arXiv:2609.34666v1cs.ROcs.LG

TL;DR

Differentiable robotic manipulation relies on gradients through long, contact-rich simulations, but their numerical reliability is not fully characterized. Using elastoplastic and granular MPM system-identification cases, this study finds that reduction order, rollout length, and observation and loss construction affect gradients and optimization outcomes.

  • Problem

    Robotic material-manipulation applications use differentiable simulation for parameter identification, demonstration generation, skill optimization, and robot design, but the numerical properties of gradients through these simulations need examination.

    Source: §I · pp. 1–2

  • Method

    The study varies precision, rollout horizon, reduction strategy, particle sampling, and geometric objective in two MPM system-identification cases while keeping contact formulation fixed.

    Source: §IV · p. 3

  • Results

    Unordered reductions changed long-horizon gradients, finite-difference checks became less reliable at longer rollouts, and observation and loss construction changed optimization behaviour and evaluation ranking.

    Source: §IV · p. 3

  • Takeaways & Limitations

    The authors recommend reproducible accumulation, checking perturbation-induced loss changes against repeated-forward variation, and reporting observation, objective, and discrete selection procedures.

    Source: §IV · p. 3

    The study is limited to a TaiChi-based differentiable MPM implementation and observations from pressing playdough and scooping sand.

    Source: §IV · p. 3

Abstract

from arXiv · show

Differentiable physics is increasingly used in robotic material manipulation for system identification, trajectory or skill optimization, demonstration generation, and robot or end-effector design. These applications depend on gradients propagated through long, contact-rich simulation rollouts. We study the numerical reliability of those gradients using two Material Point Method (MPM) system-identification benchmarks derived from elastoplastic and granular manipulation. The benchmarks provide controlled cases for three effects that also arise in broader differentiable physics-based optimization. GPU many-to-one sums whose order depends on thread scheduling changed long-horizon gradients and reversed the sign of one parameter gradient relative to a deterministic reference. Finite-difference checks became less reliable for longer rollouts because repeated-run loss variation grew much faster than the loss change produced by the tested parameter perturbations. Observation and loss definitions changed optimization behaviour and the solution preferred by an independent metric. These results motivate reproducible accumulation, finite-difference validation that compares perturbation-induced loss changes with repeated-run variation, and explicit reporting of objective construction when differentiable simulation is used for robotic optimization.

I. INTRODUCTION

Differentiable physics supports several robotic material-manipulation optimization tasks, making simulator numerical properties part of the optimization loop. This study uses elastoplastic and granular testbeds to examine those properties.

I. INTRODUCTION

  • Numerical properties of differentiable simulation sit inside optimization loops for material-parameter identification, demonstration generation, skill optimization, and robot-shape design.The applications include system identification, granular-material learning, excavation and levelling, and robot-shape optimization.Source: §I · p. 1
  • System identification, trajectory and skill optimization, and end-effector design update different variables but share backward gradient propagation through simulated interactions.Accumulation order, numerical precision, rollout horizon, and observation discretization can change the local gradient presented to the optimizer.Source: §I · p. 1
  • MPM transfers material state between Lagrangian particles and a temporary Eulerian grid through P2G, grid update and contact handling, then G2P.GPU transfers combine many thread-local contributions into shared grid or gradient entries; atomic additions can arrive in varying orders, and floating-point rounding makes sums order-dependent,.Source: §I · p. 1
  • The study examines GPU reduction order, finite-difference reliability across rollout length, and observation and loss construction using DPSI-derived elastoplastic and DDBot-derived granular cases.Earlier DPSI results reported multiple parameter solutions and disagreement between Chamfer-distance and Earth-Mover’s-distance objectives; DDBot reported exploding and fluctuating gradients and rugged loss landscapes.Source: §I · p. 2

A. Benchmark settings

The benchmarks use two material models and a fixed contact formulation to examine numerical effects. Experiments vary precision, objectives, particle density, rollout horizon, and reduction strategy.

A. Benchmark settings

  • DPSI models elastoplastic material with fixed-corotated elasticity and von Mises plasticity, while DDBot models granular material with Hencky/Saint Venant-Kirchhoff elasticity and Drucker-Prager plasticity.Both use the same projection-based signed distance field contact formulation, held fixed to isolate numerical effects.Source: A. Benchmark settings · p. 2
  • The experiments compare f32 and f64, geometric objectives, particle densities for the elastoplastic case, two granular rollout horizons, and deterministic versus ordinary GPU reductions.The settings include particle observations (PRT), point-cloud observations (PCD), and Chamfer distance (CD), Earth Mover’s distance (EMD), and height-map distance (HMD).Source: A. Benchmark settings · p. 2
  • Particle observations compare simulated particles directly with a target, whereas point-cloud observations first extract a surface representation.The geometric objectives then measure discrepancy between those observations and the target.Source: A. Benchmark settings · p. 2

B. Gradient evaluation and reproducible accumulation

Gradients are evaluated by reverse-mode automatic differentiation and central finite differences, with repeated-forward loss variation used to assess finite-difference reliability. A deterministic fixed-point alternative is compared with ordinary GPU atomic addition.

B. Gradient evaluation and reproducible accumulation

  • Reverse-mode automatic differentiation through the MPM rollout is compared with central finite differences over several perturbation sizes.For each perturbation, the loss change is compared with the spread from repeating an identical forward simulation.Source: B. Gradient evaluation and… · p. 2
  • The repeated-run loss spread indicates whether a finite-difference numerator is distinguishable from ordinary numerical variation.Long trajectories use checkpointing to limit stored simulation state.Source: B. Gradient evaluation and… · p. 2
  • The deterministic alternative maps bounded contributions to a common integer scale, adds them exactly within the allocated range, then converts the result back to floating point.It is compared with ordinary GPU atomic addition, whose contribution order can vary between runs.Source: B. Gradient evaluation and… · p. 2

A. Parallel reductions can change the reverse gradient

Unordered GPU reductions can alter long-horizon gradients, including reversing a Poisson-ratio gradient’s sign against a deterministic reference. Deterministic reductions produced identical state and loss checksums in five selected repeat cases.

A. Parallel reductions can change the reverse gradient

  • On the 311-step DDBot-derived case, the GPU ensemble mean for ∂L/∂ν was +1.29, versus −7.99 for the deterministic single-threaded CPU reference.The two signs imply opposite updates of Poisson’s ratio; repeated runs first diverged at P2G, with small differences propagating through later simulation and reverse-mode operations.Source: A. Parallel reductions can change the… · p. 2
  • Deterministic integer reductions and fixed-order summation of repeated Chamfer-gradient contributions yielded identical state and loss checksums in all five representative configurations.The selected cases included f32 and f64 and both benchmark families.Source: A. Parallel reductions can change the… · p. 2

B. Finite-difference checks can be unreliable for longer rollouts

Finite-difference checks become less reliable as rollouts lengthen: repeated-run loss variation grows faster than the tested perturbations’ loss changes. At longer horizons, f64 estimates also stop converging for the worst component, while validation costs more.

B. Finite-difference checks can be unreliable for longer rollouts

  • At two steps, perturbation-induced loss changes were 93–806 times the measured f32 loss spread, but at ten or more steps the ratio fell to 2–14.Repeated-run loss variation increased much faster than the loss change caused by the tested perturbations.Source: B. Finite-difference checks can be… · p. 2
  • At 20 and 40 steps, the worst f64 AD/FD discrepancies rose to 3.41×10−1 and 2.70×10−1, and the finite-difference sequence no longer converged for the worst component.At two steps, the worst relative discrepancy was at most 1.5×10−5; it remained below 3 × 10−5 at five and ten steps.Source: B. Finite-difference checks can be… · p. 2
  • Fig. 2b shows that finite-difference agreement in the DPSI-derived case depends on rollout horizon.Validation cost increased from 25 s to 178 s, so longer rollouts did not provide a stronger finite-difference reference in this experiment.Source: B. Finite-difference checks can be… · p. 2, Figure 2 · p. 3

C. Observation and loss design change optimization

Observation and loss choices affect both descent frequency and which objective an independent metric favors. Their construction also shapes the optimization signal and, in system identification, can change recovered material parameters.

C. Observation and loss design change optimization

  • PRT-CD reduced its objective in 78 of 80 attempted epochs (97.5%), versus 67.5% for PRT-EMD, 53.8% for PCD-CD, and 57.5% for PCD-EMD.These rates were measured under the common numerical setup used here.Source: C. Observation and loss design change… · p. 3
  • Mean terminal HMD was 3738.4 mm for PRT-CD and 3698.6 mm for PRT-EMD, reversing the ranking suggested by PRT-CD’s more frequent descent.Increasing sampling from 218 to 876 particles raised occupied surface cells from 89 to 151, a factor of 1.70 rather than four; in one 876-particle case, f32 and f64 selected different surface points and changed PCD-EMD by 0.517%.Source: C. Observation and loss design change… · p. 3
  • The observation pipeline as well as simulated physics contributes to the loss, and its choices can change recovered material parameters in system identification.For trajectory, policy, or design optimization, these choices can change the gradient direction associated with controls, contact sequences, or shape variables.Source: C. Observation and loss design change… · p. 3

IV. CONCLUSION

In two MPM system-identification testbeds, unordered reductions changed long-horizon gradients, finite-difference checks weakened with rollout length, and observation and loss choices altered optimization and evaluation ranking. The authors recommend reproducible accumulation, variation-aware finite-difference checks, and explicit objective reporting.

IV. CONCLUSION

  • Small unordered-reduction errors produced long-horizon gradient changes, including one Poisson-ratio sign reversal; deterministic reductions made five selected repeat cases bit-identical.The study also found that finite-difference loss-change-to-run-variation ratios fell from 93–806 at two steps to 2–14 at horizons of ten steps or more, while objective choices changed optimization and independent-HMD ranking.Source: §IV · p. 3
  • Longer rollouts weakened finite-difference checks, and objective choices changed optimization and independent-HMD ranking; the authors recommend reproducible sums, variation-aware checks, and explicit objective reporting.Evidence is limited to TaiChi-based differentiable MPM and two motions: playdough pressing and sand scooping.Source: §IV · p. 3

A. One MPM substep

  • MPM transfers particle contributions to a shared grid, updates the grid and handles contact, then interpolates velocity back to advance particle deformation and position.Source: A. One MPM substep · p. 5

B. Material and contact models

  • The testbeds use fixed-corotated elasticity with von Mises plasticity and Hencky/Saint Venant-Kirchhoff elasticity with Drucker-Prager plasticity, under shared SDF contact.Source: B. Material and contact models · p. 5

C. Observation operators and losses

  • EMD pairs points one-to-one, recording unmatched targets when counts differ; HMD sums per-cell height differences, while PCD surface-winner choices can change discretely.Source: C. Observation operators and losses · p. 5

A. Reverse-mode computation in operational form

  • Checkpointed reverse-mode recomputes forward states per window, reverses simulator operations, and inserts a hand-written SVD VJP; finite-difference checks compare loss changes with repeat-run spread.Source: A. Reverse-mode computation in… · p. 6

B. Parallel reductions in MPM

  • GPU atomic additions allow scheduler-dependent accumulation order; floating-point rounding can make those orders differ, first diverging in P2G and also affecting reverse-mode sums.Source: B. Parallel reductions in MPM · p. 6

C. Fixed-point integer accumulation

  • Fixed-point accumulation quantizes contributions to a common scale and sums integers independently of arrival order; the 27,440-particle DDBot P2G reduction retained 23 bits beyond f64 precision.Source: C. Fixed-point integer accumulation · pp. 6–7

APPENDIX C EXPERIMENTAL PROTOCOL

The protocol spans 64 primary experiment cells across DPSI-derived and DDBot-derived cases, with five representative cells repeated for reproducibility. In the horizon experiment, longer rollouts made finite-difference validation less reliable as repeated-run loss variation outpaced perturbation-induced loss changes.

APPENDIX C EXPERIMENTAL PROTOCOL

  • The 64 primary cells comprise 48 DPSI-derived and 16 DDBot-derived configurations, plus five repeated cells for 69 campaign tasks.DPSI varies four objectives, precision, particle density, and two seeds at a 94-step horizon; DDBot varies two objectives, precision, horizon, and two seeds.Source: Appendix C · p. 7, Table I · p. 7
  • Across base DDBot-derived cells, 14 of 58 accepted f32 steps and 13 of 42 accepted f64 steps selected the opposite direction.RMSProp was followed by line-search evaluation of candidate scales in both gradient directions.Source: Appendix C · p. 7
  • The DPSI-derived density settings use 218, 451, and 876 particles, producing 89, 105, and 151 occupied PCD surface cells.Added particles often project into cells that are already occupied.Source: Appendix C · p. 7

A. Gradient validation across horizon

  • At two steps, ΔLθ(h)/V ranged from 93 to 806, but fell to 2–14 at horizons of ten steps or more.Repeated-run f32 loss spread increased faster than the perturbation-induced loss change; at 20 and 40 steps, the tested relative perturbations also failed to produce a converged f64 sequence for the worst component.Source: A. Gradient validation across horizon · p. 7, Table II · p. 7

B. Objective and particle-density results

The objective and particle-density experiments show that optimization outcomes depend on the observation objective and sampling density. The section also reports computational measurements and relates the controlled tests to earlier DPSI and DDBot findings.

B. Objective and particle-density results

  • At f32, PRT-EMD total loss change shifted from −0.87%/−0.48% at reference density to −2.60%/−1.26% at 2× and −5.21%/−4.06% at 4×.Values are for seeds 0/1.Source: B. Objective and particle-density… · p. 7
  • Changes went in the other direction, from −10.75%/−2.80% at reference density to −0.64%/−0.30% at 2× and −0.07%/−0.60% at 4×.Source: B. Objective and particle-density… · p. 8
  • Particle count rose from 218 to 876 while occupied PCD surface cells increased only from 89 to 151 on the fixed 32 × 32 grid.Between the first two density levels, 93% of newly added particles landed in already occupied cells; between the second and third, the figure was 89%.Source: B. Objective and particle-density… · p. 8

C. Runtime and reproducibility measurements

  • At 4,000 particles, the contact-backend speedup ranged from 35.1× to 92.9×, depending on precision and device.The GPU version removed six host/device frame transfers per MPM substep; the 94-step benchmark gave an end-to-end 12.73× value.Source: C. Runtime and reproducibility… · p. 8
  • For f32 grid mass, deterministic accumulation had maximum error 2.00 × 10^-10 against the fixed-order long-double reference, versus 1.44 × 10^-9 for the best of eight atomic runs.The fixed-point representation provided order-independent repetition in the tested reductions.Source: C. Runtime and reproducibility… · p. 8

D. Connection to earlier DPSI and DDBot observations

  • The controlled objective measurements extend earlier DPSI reports of CD/EMD disagreement and DDBot reports of fluctuating long-trajectory gradients.The DDBot-derived tests isolate unordered GPU reductions as one numerical source of variability.Source: D. Connection to earlier DPSI and… · p. 8
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