Iterations and Nesterov momentum

23 · Convergence

A stack of eight 10 cm boxes. Each Gauss-Seidel sweep carries load only a short way down the stack, so the bottom of a tall stack converges last and the top sags when there are few iterations. Nesterov momentum with restart speeds up exactly this slow, low-frequency error.

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Cost

Time per step, CUDA graph replay, median of 7 interleaved rounds on one NVIDIA RTX PRO 6000 Blackwell Server Edition; ratios are against the first row. Solver allocation is the GPU memory the solver allocates at construction, before warm-up and graph capture. Scene: the 8-box stack in each world, 8 steps of 1/240 s per frame.

Option	1 world	64 worlds	1024 worlds	solver allocation, 1024 worlds	result in this tutorial
iterations=6	0.67 ms	0.85 ms	1.24 ms	215 MB	sag 4.6 mm
iterations=6 + Nesterov	0.73 ms 1.09×	0.91 ms 1.07×	1.30 ms 1.05×	219 MB	sag 3.0 mm
iterations=12	1.01 ms 1.51×	1.21 ms 1.42×	1.65 ms 1.33×	215 MB	sag 2.1 mm
iterations=12 + Nesterov	1.10 ms 1.64×	1.30 ms 1.53×	1.88 ms 1.51×	219 MB	sag 0.63 mm
iterations=24	1.67 ms 2.48×	1.87 ms 2.20×	2.53 ms 2.04×	215 MB

What to look at

6 iterations: the top sags about 4.6 mm, and 3 mm with Nesterov.
12 iterations: about 2 mm; with Nesterov about 0.6 mm.

Key parameter

iterations / nesterov_momentum

PGS iterations per step (default 12). nesterov_momentum adds accelerated sweeps with restart (default off).

Try changing

Stack 16 boxes.
Try iterations=24 without momentum.

Concept

Nesterov momentum

What it is

Gauss-Seidel removes high-frequency error quickly and low-frequency error slowly; a tall stack's load path is the slow kind. Nesterov momentum starts each sweep from impulses extrapolated along the previous change, and restarts the momentum whenever a sweep's correction opposes the extrapolation (O'Donoghue-Candès restart), as MuJoCo's PGS does.

y = λₖ + βₖ (λₖ − λₖ₋₁), βₖ = (k−1)/(k+2)

In this scene

Top sag falls from 4.6 to 3.0 mm at 6 iterations and from 2.1 to 0.63 mm at 12.

Considerations

Cost: +5 to 13% per step (12 iterations, 1024 worlds: 1.65 → 1.88 ms) and 8 bytes per row of history (4.5 MB at 1024 worlds).
Interactions: Requires serial solve, immediate response and interleaved rows; bypasses the specialized sparse and local solves. CUDA only.
Use it when: Stacks, mass ratios and long chains at 8+ iterations; gains are small at very low counts.

The script

cd tutorials && python 23_iterations_nesterov.py prints each variant's final metric; --record out.json also writes the replay. Every tutorial imports replay.py from the same folder.

23_iterations_nesterov.py
replay.py

# SPDX-FileCopyrightText: Copyright (c) 2026 Skild AI and the Steel contributors
# SPDX-License-Identifier: Apache-2.0

"""Iterations and Nesterov momentum on a tall stack."""

import newton
import warp as wp
from newton.solvers import SolverSteel

from replay import Recorder

DT = 1.0 / 240.0
SUBSTEPS = 8
N, H = 8, 0.05  # eight 10 cm boxes


def build():
    builder = newton.ModelBuilder()
    builder.add_ground_plane(cfg=newton.ModelBuilder.ShapeConfig(mu=0.6))
    for i in range(N):
        box = builder.add_body(
            xform=wp.transform((0.0, 0.0, H + 2 * H * i + 0.0002 * i), wp.quat_identity())
        )
        builder.add_shape_box(box, hx=H, hy=H, hz=H, cfg=newton.ModelBuilder.ShapeConfig(mu=0.6))
    return builder.finalize()


def simulate(rec, label, **options):
    model = build()
    solver = SolverSteel(model, **options)
    pipeline = newton.CollisionPipeline(model)
    contacts = pipeline.contacts()
    state_0, state_1, control = model.state(), model.state(), model.control()

    run = rec.variant(label, model)
    for _ in range(rec.frames(4.0)):
        top = state_0.body_q.numpy()[N - 1, 2]
        run.frame(state_0, sag_mm=1000.0 * (H * (2 * N - 1) - top))
        for _ in range(SUBSTEPS):
            pipeline.collide(state_0, contacts)
            solver.step(state_0, state_1, control, contacts, DT)
            state_0, state_1 = state_1, state_0
    run.check(solver)


if __name__ == "__main__":
    rec = Recorder()
    simulate(rec, "iterations=6", iterations=6)
    simulate(rec, "iterations=6, nesterov_momentum=True", iterations=6, nesterov_momentum=True)
    simulate(rec, "iterations=12 (default)")
    simulate(rec, "iterations=12, nesterov_momentum=True", nesterov_momentum=True)
    rec.save()

# SPDX-FileCopyrightText: Copyright (c) 2026 Skild AI and the Steel contributors
# SPDX-License-Identifier: Apache-2.0

"""Record Steel tutorial runs as replay JSON for the tutorials page, or print a summary.

Every tutorial imports this module. ``python NN_name.py`` runs the variants and prints
the final metric of each; ``--record out.json`` also writes body poses, shapes and metric
series that the page replays in 3D.
"""

import argparse
import base64
import json
import os
import subprocess
import sys
import time

import newton
import numpy as np
import warp as wp

from source_hash import source_sha256

_RENDERED_TYPES = {
    newton.GeoType.PLANE,
    newton.GeoType.SPHERE,
    newton.GeoType.CAPSULE,
    newton.GeoType.ELLIPSOID,
    newton.GeoType.CYLINDER,
    newton.GeoType.BOX,
    newton.GeoType.MESH,
    newton.GeoType.CONE,
    newton.GeoType.CONVEX_MESH,
}


class Recorder:
    """Collect the variants of one tutorial; ``fps`` is the replay frame rate."""

    def __init__(self, fps: int = 30):
        parser = argparse.ArgumentParser()
        parser.add_argument(
            "--record", metavar="OUT.json", help="write a replay file for the tutorials page"
        )
        parser.add_argument("--device", default=None, help="Warp device, for example cuda:0")
        self.args = parser.parse_args()
        if self.args.device:
            wp.set_device(self.args.device)
        self.fps = fps
        self.script = os.path.abspath(sys.argv[0])
        self.variants: list[Variant] = []

    def frames(self, seconds: float) -> int:
        """Number of replay frames in ``seconds``, including the first."""
        return int(round(seconds * self.fps)) + 1

    def variant(
        self,
        label: str,
        model: newton.Model,
        *,
        ghost: bool = False,
        slot: int | None = None,
        scene: str | None = None,
    ) -> "Variant":
        """Start one run; ``ghost`` overlays run ``slot``, ``scene`` groups runs for charts."""
        variant = Variant(label, model, self.fps, ghost, slot, scene)
        self.variants.append(variant)
        return variant

    def save(self) -> None:
        """Print a summary and, with ``--record``, write the replay file."""
        for variant in self.variants:
            last = variant.metrics[-1] if variant.metrics else {}
            flag = "  (rows dropped)" if variant.overflow else ""
            print(f"{variant.label:<48} {json.dumps(last)}  {variant.wall:.2f} s{flag}")
        if not self.args.record:
            return
        source_sha = source_sha256(self.script)
        doc = {
            "script": os.path.basename(self.script),
            "script_sha256": source_sha,
            "newton_commit": _newton_commit(),
            "newton_version": newton.__version__,
            "warp_version": wp.config.version,
            "gpu": wp.get_device().name,
            "fps": self.fps,
            "variants": [v.to_json() for v in self.variants],
        }
        with open(self.args.record, "w") as f:
            json.dump(doc, f, separators=(",", ":"))
        print(f"wrote {self.args.record}")


class Variant:
    """One recorded run: a pose per frame for every body, plus named metric values."""

    def __init__(self, label, model, fps, ghost, slot, scene):
        self.label, self.model, self.fps, self.ghost, self.slot = label, model, fps, ghost, slot
        self.scene = scene
        self.poses: list[np.ndarray] = []
        self.metrics: list[dict] = []
        self.awake: list[list[int]] = []
        self.particles: list[np.ndarray] = []
        self.overflow = False
        self.start = time.perf_counter()
        self.wall = 0.0

    def frame(self, state: newton.State, awake=None, particles=None, **metrics) -> None:
        """Record ``state.body_q``, optional awake flags, particle positions and metric values."""
        self.poses.append(state.body_q.numpy().astype(np.float32).copy())
        if particles is not None:
            self.particles.append(particles.numpy().astype(np.float32).copy())
        self.metrics.append({k: _round(v) for k, v in metrics.items()})
        if awake is not None:
            self.awake.append([int(x) for x in awake.numpy()])
        self.wall = time.perf_counter() - self.start

    def check(self, solver) -> None:
        """Note whether the solver dropped rows; call once after the run."""
        try:
            solver.check_constraint_capacity()
        except RuntimeError:
            self.overflow = True

    def to_json(self) -> dict:
        poses = (
            np.stack(self.poses)
            if self.poses
            else np.zeros((0, self.model.body_count, 7), np.float32)
        )
        return {
            "label": self.label,
            "fps": self.fps,
            "ghost": self.ghost,
            "slot": self.slot,
            "scene": self.scene,
            "bodies": int(self.model.body_count),
            "frames": int(poses.shape[0]),
            "poses": base64.b64encode(poses.astype(np.float32).tobytes()).decode(),
            "shapes": _shape_table(self.model),
            "metric": self.metrics,
            "awake": self.awake or None,
            "particles": self._particles_json(),
            "wall_s": round(self.wall, 3),
            "overflow": self.overflow,
        }

    def _particles_json(self):
        if not self.particles:
            return None
        frames = np.stack(self.particles)
        tris = self.model.tri_indices
        radius = self.model.particle_radius.numpy()
        return {
            "count": int(frames.shape[1]),
            "frames": base64.b64encode(frames.astype(np.float32).tobytes()).decode(),
            "radius": round(float(radius.mean()), 5) if radius.size else 0.0,
            "tris": [int(v) for v in tris.numpy().reshape(-1)]
            if tris is not None and tris.shape[0]
            else None,
        }


def _shape_table(model):
    types = model.shape_type.numpy()
    scales = model.shape_scale.numpy()
    xforms = model.shape_transform.numpy()
    bodies = model.shape_body.numpy()
    flags = model.shape_flags.numpy()
    shapes = []
    for i, t in enumerate(types):
        if int(t) not in _RENDERED_TYPES or not flags[i] & int(newton.ShapeFlags.VISIBLE):
            continue
        shape = {
            "b": int(bodies[i]),
            "t": int(t),
            "s": [round(float(v), 5) for v in scales[i]],
            "x": [round(float(v), 6) for v in xforms[i]],
        }
        if int(t) in (newton.GeoType.MESH, newton.GeoType.CONVEX_MESH):
            mesh = model.shape_source[i]
            shape["v"] = [round(float(v), 5) for v in np.asarray(mesh.vertices).reshape(-1)]
            shape["i"] = [int(v) for v in np.asarray(mesh.indices).reshape(-1)]
        shapes.append(shape)
    return shapes


def _round(value):
    value = float(value)
    return round(value, 5) if np.isfinite(value) else None


def _newton_commit():
    if os.environ.get("NEWTON_COMMIT"):
        return os.environ["NEWTON_COMMIT"]
    try:
        root = os.path.dirname(os.path.dirname(newton.__file__))
        return subprocess.check_output(["git", "-C", root, "rev-parse", "HEAD"], text=True).strip()
    except (OSError, subprocess.CalledProcessError):
        return None

Recorded with SolverSteel on Newton 1.9.0.dev0, Warp 1.19.0, NVIDIA RTX PRO 6000 Blackwell Server Edition. Replays read the state back every frame, so they are not a speed benchmark. Script · page text

Steel docs and tutorials are licensed under Apache-2.0. Third-party notices. "Skild", "Skild AI" and the Skild logo are trademarks of Skild AI and are not licensed under Apache-2.0.