Pratyush-01/physix-live-src

PhysiX — Equation Discovery via RLVR

An OpenEnv hackathon submission (Apr 2026).

Given a noisy trajectory and a one-sentence hint, a language model iteratively proposes and refines an ODE that reproduces the observed motion. Reward comes entirely from scipy.integrate.odeint + per-step R² — no LLM-as-judge.

---

Links

Live demo (HF Space)	https://huggingface.co/spaces/Pratyush-01/physix-live
Trained model	https://huggingface.co/Pratyush-01/physix-3b-rl
Colab training notebook	train/physix_train_colab.ipynb
W&B training runs	https://wandb.ai/pratyush01/physix-live
Writeup	docs/writeup.md

---

Environment

Physical Systems — 3 Tiers

6 training systems across 3 difficulty tiers, plus 2 held-out Tier 3 systems.

Tier	System	Ground-truth equation	Notes
1	Free Fall	d2y/dt2 = -g	1 parameter
1	Free Fall with Drag	d2y/dt2 = -g + k*vy**2	nonlinear drag
1	Simple Pendulum	d2theta/dt2 = -(g/L)*sin(theta)	transcendental
2	Damped Pendulum	d2theta/dt2 = -(g/L)*sin(theta) - b*dtheta	3 parameters
2	Spring-Mass	d2x/dt2 = -(k/m)*x	parameter ratio
2	Damped Spring	d2x/dt2 = -(k/m)*x - (b/m)*dx	damped oscillation
3 (held-out)	Projectile with Drag	2-D coupled ODE	out-of-distribution
3 (held-out)	Charged Particle in B-field	2-D Lorentz force	cross-product coupling

Parameters and initial conditions are randomised per episode.

Example Task

HINT: Object dropped from altitude 58.3 m, mass 1.8 kg, in air.
      Air resistance may be non-negligible.

TRAJECTORY (t, y, vy):
  t=0.00  y=58.30  vy= 0.00
  t=0.50  y=56.89  vy=-5.44
  t=1.00  y=51.91  vy=-9.21
  t=2.00  y=35.70  vy=-13.88
  t=3.00  y=16.11  vy=-16.02
  t=5.00  y=-20.42 vy=-16.49   ← terminal velocity visible

STATS: mean_vy=-10.85  std_vy=6.41  min_vy=-16.49

Target output:

{
  "equation": "d2y/dt2 = -g + k * vy**2",
  "params": {"g": 9.81, "k": 0.047},
  "rationale": "Quadratic drag; vy² is positive regardless of sign of vy."
}

Grammar: operators + - * / **, functions sin cos tan exp log sqrt abs, declared state variables and parameter names. Anything outside this scores format = 0.

Episode Flow

sequenceDiagram
    participant Agent
    participant Env as PhysiXEnvironment
    participant Sim as scipy.odeint
    participant Verifier

    Env->>Agent: reset() → trajectory + hint
    loop up to 8 turns
        Agent->>Env: step(equation + params + rationale)
        Env->>Sim: simulate hypothesis from t=0
        Sim-->>Verifier: predicted trajectory
        Verifier-->>Env: r_match + r_progress + r_simplicity + r_format
        Env->>Agent: mismatch summary + reward breakdown + history
        alt r_match > 0.93 or budget exhausted
            Env-->>Agent: done=True
        end
    end

After each step the agent receives an English mismatch summary (e.g. "predicted vy diverges after t=2 s; residual consistently negative") alongside the numeric reward breakdown, so it has something to act on in the next turn.

---

Reward

All reward is computed from scipy.odeint output — no model-in-the-loop scoring.

Step-wise (live env + GRPO)

Component	Weight	Formula	Purpose
match	0.50	R² (observed vs. predicted)	primary accuracy signal
progress	0.20	max(0, r_match − r_match_prev)	per-turn improvement shaping
simplicity	0.20	1 − (operator_count / 12)	prefer shorter equations
format	0.10	1 if parsed and simulated successfully	syntactic + numerical validity

GRPO-only additions

Two extra signals are added during training but not used in the live env:

• match_dense = sqrt(R²) — gives a non-trivial gradient when raw R² is near zero (e.g. sqrt(0.05) ≈ 0.22).
• correctness = 1 if R² ≥ 0.70, else 0 — a binary bonus that helps push past R² plateaus where the dense signal flattens.

Reward-hacking mitigations

Three failure modes found during development and how they were closed:

1. Parse-but-crash exploit. A valid-but-explosive equation (e.g. d2y/dt2 = exp(vy**10)) parses but makes odeint produce NaN. Without a fix, it earns format = 1.
→ format = 1 only if integration completes without NaN/inf.

2. Trivial-equation exploit. d2y/dt2 = 0 has zero operators, so simplicity = 1, earning 20% reward for a completely wrong trajectory.
→ simplicity = 0 unless r_match ≥ 0.10.

3. Progress signal in single-turn GRPO. Every GRPO training row starts with previous_r_match = 0, so progress = r_match — a redundant copy of the match signal that dilutes advantage estimates.
→ progress is excluded from the GRPO reward function set; it is only used in multi-turn live episodes.

---

Training: SFT → GRPO

Why SFT first

GRPO relies on reward variance across rollouts to estimate advantages. With a cold base model, ~80% of completions are unparseable (LaTeX, prose, malformed JSON) and most parseable ones crash the integrator, leaving near-zero variance and no useful gradient. The model needs to produce the right output format before RL can do anything meaningful with the physics signal.

SFT runs for 3 epochs on synthetic (prompt, ground_truth_equation) pairs generated from the environment. After SFT:

• 90% of completions parse and simulate successfully (up from ~20%).

• Equations are in the ASCII ODE grammar the verifier expects.
• The model has seen the right equation family for each system at least once.

SFT only establishes format. Parameter values are still wrong — that is what GRPO refines.

Step 1 — SFT warm-start

python -m physix.training.sft \
  --model Qwen/Qwen2.5-3B-Instruct \
  --output-dir runs/physix-3b-sft \
  --epochs 3 \
  --lora-r 32 \
  --instances-per-system 32 \
  --system-ids damped_spring

Runtime: ~5 min on L40S.

Step 2 — GRPO

python -m physix.training.loop \
  --model Qwen/Qwen2.5-3B-Instruct \
  --output-dir runs/physix-3b-rl \
  --num-steps 200 \
  --num-generations 4 \
  --lora-r 32 \
  --sft-checkpoint runs/physix-3b-sft/merged \
  --system-ids damped_spring \
  --push-to-hub \
  --hub-repo-id Pratyush-01/physix-3b-rl

Runtime: ~45 min on L40S.

Full cloud job

hf jobs uv run train/job_train_single.py \
    --image unsloth/unsloth:2026.3.8-pt2.9.0-vllm-0.16.0-cu12.8-studio-release \
    --flavor l40sx1 \
    --secrets HF_TOKEN \
    --secrets WANDB_API_KEY \
    -v hf://datasets/Pratyush-01/physix-live-src:/physix-live \
    --timeout 2h

---

Training Results

GRPO Loss (↓)	Total Reward (↑)

Per-component reward breakdown

W&B runs: pratyush01/physix-live

Key observations from the run:

• reward/format rises quickly and stabilises near 0.9+ — the SFT warm-start does its job.
• reward/match and reward/match_dense climb gradually through GRPO; neither saturates by step 200, suggesting more steps would help.
• reward/simplicity stays roughly stable — the model isn't gaming it, which is what the gate at R²=0.10 is for.

What we don't claim:

• The model generalises well to Tier 3 systems without further training.
• A 3B model is competitive with frontier models on this task.
• The reward improvements translate to meaningful physics understanding beyond the training distribution.

---

Repository Layout

physix-live/
├── physix/
│   ├── models.py                 # Pydantic Action / Observation / State
│   ├── client.py                 # OpenEnv WebSocket client
│   ├── systems/                  # physical systems
│   │   ├── base.py               # PhysicalSystem ABC
│   │   ├── tier1.py              # FreeFall, FreeFallWithDrag, SimplePendulum
│   │   ├── tier2.py              # DampedPendulum, SpringMass, DampedSpring
│   │   ├── tier3.py              # ProjectileWithDrag, ChargedInBField (held out)
│   │   └── registry.py
│   ├── verifier/
│   │   ├── parser.py             # SymPy whitelisted grammar
│   │   ├── simulator.py          # scipy.odeint forward simulation
│   │   ├── metrics.py            # per-step R²
│   │   ├── mismatch.py           # English residual summary
│   │   └── reward.py             # reward composition + hacking mitigations
│   ├── server/
│   │   ├── environment.py        # PhysiXEnvironment (OpenEnv subclass)
│   │   ├── interactive.py        # session-based REST router for the UI
│   │   └── app.py
│   └── training/
│       ├── prompt.py             # observation → prompt
│       ├── scorer.py             # cached single-completion scorer
│       ├── reward_fns.py         # TRL-compatible reward callables
│       ├── dataset.py            # GRPO dataset builder
│       ├── sft.py                # SFT warm-start
│       └── loop.py               # Unsloth + TRL GRPO loop
├── frontend/                     # React + TS + Tailwind demo UI
├── train/                        # HF Jobs launcher + Colab notebook
│   ├── submit.py                 # submit job via HfApi.run_uv_job
│   ├── job_train.py              # multi-system driver (in-container)
│   ├── job_train_single.py       # single-system driver (in-container)
│   ├── physix_train_colab.ipynb  # SFT → GRPO end-to-end notebook
│   └── sync-plots.sh             # mirror plots from model repo
├── tests/                        # ~30 tests
├── docs/
│   ├── plots/                    # committed loss / reward / per-component PNGs
│   └── writeup.md
├── Dockerfile                    # env Space build (FastAPI + built React UI)
├── openenv.yaml                  # OpenEnv manifest (name, runtime, app entrypoint)
└── pyproject.toml

---

Quick Start

One command from the repo root:

make dev

This starts the FastAPI backend on :8000 (deps auto-resolved by uv) and the Vite frontend on :5173. Open http://localhost:5173.

Connecting an LLM

The demo speaks to any OpenAI-compatible /v1/chat/completions endpoint — local Ollama, Hugging Face Inference Providers, OpenAI, vLLM, OpenRouter, etc. The "Connect an LLM" panel exposes:

Field	Purpose
Endpoint	Preset dropdown. Picks base_url + a default model id.
Model	Provider-native id (HF repo, Ollama tag, OpenAI name). Free-form.
Custom base URL	Shown when Custom is selected. Anything ending in /v1.
API key	Bearer token. Persisted per base_url in localStorage, never sent unless an episode runs.

Server-side env-var fallback (lets a deployed Space ship a sensible default without leaking secrets in the bundle):

URL family	Env var
*huggingface*	HF_TOKEN, then HUGGINGFACE_API_KEY
*openai.com*	OPENAI_API_KEY
*openrouter*	OPENROUTER_API_KEY
localhost / 127.0.0.1	none (Ollama needs no key)

Side-by-side comparison

The default page is a two-column comparison: same trajectory, same hint, same seed, same verifier — two different models. The presets are wired to make the headline story self-evident:

• A = Pratyush-01/physix-3b-rl via HF Inference Providers (the GRPO-trained model)
• B = Qwen/Qwen2.5-3B-Instruct via HF Inference Providers (untrained baseline)

Drop in gpt-4o-mini on either side as a frontier reference, or swap to local Ollama for offline dev. The reward delta between the two columns is exactly what GRPO bought — no benchmark-prose necessary.

For the trained model on HF Inference Providers: weights are public, but the repo card needs inference: true and a serving provider (Featherless/Together/etc.) to have it loaded. If a visitor sees a 404 from the trained side, they can either bring up ollama serve locally and pull a quantised version, or fall back to Qwen/Qwen2.5-3B-Instruct on both sides.

Programmatic use

import asyncio
from physix import PhysiXEnv, PhysiXAction

async def main():
    async with PhysiXEnv(base_url="http://127.0.0.1:8000") as env:
        obs = await env.reset(system_id="free_fall_drag", seed=42)
        result = await env.step(
            PhysiXAction(equation="d2y/dt2 = -g + k * vy**2", params={"g": 9.81, "k": 0.05})
        )
        print(result.observation.reward_breakdown)

asyncio.run(main())

pytest tests/

---

License

MIT.