Applications: Structure-Preserving PINNs (Paper 1) — framework + inverse/traffic study

New top-level applications/ area (research papers built on mononet, out of the wheel) and its first occupant, applications/pinn: structure-preserving PINNs, where mononet's expressive hard monotonicity enforces a PDE solution's admissibility (TVD/entropy) by construction instead of via soft penalties.

Draft — headline (both tiers), the residual-vs-plain field ablation, and the high-noise robustness sweep (both fields) are in. Pending: figures, a non-PINN/smoothness-prior baseline (being added), cross-backend check.

Central result: robustness under noise (two PDEs, sparsity × noise, 10-seed IQM)

Inverse reconstruction from sparse noisy observations, on LWR (concave flux) and Burgers (convex flux).

Whole-field accuracy (L¹) — the general result: from noise ≥ 0.05, hard-monotone has the lowest L¹ error in 16/16 noisy LWR cells and 15/16 noisy Burgers cells (all observation counts), margin widening with noise (n_obs=80/0.20: LWR 13.2 vs 17.0/21.5; Burgers 19.0 vs 20.6/29.3). And it is exactly admissible (violation 0) at every one of the 40 cells vs baselines oscillating to 0.9–1.3. The monotone prior regularizes the noise the baselines over-fit everywhere.

Front-weighted accuracy (L²) is PDE-dependent: clean on LWR (hard best from 0.05, e.g. 0.20: 1.39 vs 1.59/1.97) but mixed on Burgers — its steeper shock lets the oscillating baselines match/beat hard on L² at high noise (0.20: 1.90 vs 1.82, while they violate 0.86). L² is dominated by the sharp front, where a continuous monotone ramp can't beat a steep (oscillating) fit; L¹ integrates the whole field, where the prior wins broadly.

Honest summary: robust general claims are (i) lower whole-field (L¹) error under noise on both PDEs and (ii) guaranteed admissibility everywhere; the "beats on every metric" reading holds cleanly only for LWR (Burgers L² is a wash-to-slight-loss at the highest noise). We report both norms rather than lead with the flattering one.

Headline panels (MonoResidual, 10-seed IQM, equal-budget tuned)

Inverse flagship (noise 0.02): hard-monotone L¹ 3.52 / L² 0.714 / viol 0 — best + only admissible; vanilla 3.92/0.731/0.083; soft 3.95/0.723/0.104; weight_clip 22.5/2.10 (fails → the effect is expressiveness, not monotonicity per se). Forward tier (worst case): constraint pays an accuracy cost (L¹ 8.1 vs 6.1), stays oscillation-free — a mechanism check.

Result strength & relevance (honest assessment)

Strong:

• General & robust: crossover across 4 sparsities × 5 noise levels × both fields (20 cells, monotone trend) — a real mechanism, not a single-point fluke.
• Rigorous: equal-budget Optuna per method (baselines not under-tuned), 10-seed IQM, tuned-once/stress-tested, fully reproducible (artifacts + runners + RUNBOOK).
• Interpretable + controlled: prior helps exactly when data is scarce/noisy; the soft baseline (tuned monotonicity penalty) still loses and the inexpressive weight_clip fails → it is specifically expressive hard monotonicity.
• Dual benefit: lower error and guaranteed admissibility — the latter has independent value for trust/safety that L² undersells.
• Non-PINN comparator (new): a tuned classical RBF smoother (no PDE, no monotonicity) is ~2× worse across all noise (n_obs=80: 1.37→2.19 vs 0.70→1.26) and inadmissible — so the win needs both the PDE and expressive hard monotonicity; generic smoothing reproduces neither.

Threats to validity / scope:

• 1-D, single PDE (LWR), monotone-solution class — the prior is known-true here by construction; no evidence yet where monotonicity is only approximately right, or in higher dimensions.
• Synthetic i.i.d. Gaussian noise favors a smoothing prior; real sensor noise is biased/correlated/MNAR.
• Monotonicity-vs-generic-smoothness (within-PINN) not fully separated — a curvature-penalty PINN would isolate it given the PDE. (A non-PINN smoothness comparator is now included — see below.)
• Magnitude moderate (~20–36% at high noise); the sweep reports IQM (robustness comes from the cross-cell trend, not per-cell CIs).

Relevance: high for the paper's thesis (turns "admissible but not more accurate" into "better accuracy + guaranteed structure, growing with noise"); moderate for the broader field, bounded by the 1-D/single-PDE/monotone-class scope. Lift items, in priority: (1) a 2nd PDE or higher-D, (2) a real noisy dataset, (3) a within-PINN curvature-penalty control.

What's here

Framework (admissibility abstraction, problem registry, NumPy core/, JAX+PyTorch builders/trainers with grad-clip + input normalization, tier-aware Optuna search, resume-able IQM runner/sweep); field ablation as a config knob; two tiers.

Upstream fix (#100)

MonoResidual had collapsed to a near-linear map; PR #100 fixes it (merged). Both backends use MonoResidual + input normalization.

Testing

uv run pytest tests/applications → 81 passed (Torch tests skip in gpu-jax); ruff + mypy clean (203 files); docs build passes.

Known gaps (RUNBOOK/FINDINGS)

Figures (crossover curve + reconstructed field); cross-backend equivalence (needs Torch → default); physical-bounds [0, ρ_max] metric.

Figures

Eight figures in paper/figures/, embedded in §6.2.1, regenerated by
python -m applications.pinn.experiments.figures:

• Reconstruction slices (both PDEs, stress point n_obs=80/noise=0.20): the money-shot — hard-monotone is a clean in-range monotone ramp while vanilla/soft overshoot and dip below zero (unphysical).
• Formats: each figure ships as vector PDF (for LaTeX \includegraphics, fonts embedded Type-42; via Git LFS) and PNG (for this markdown/Sphinx preview).
• Crossover curves (L¹/L²/admissibility vs noise): comparable at zero noise, hard-monotone L¹ separates cleanly below both baselines as noise grows; admissibility stays flat at 0.

Also merged latest origin/main (legacy port, Stage-2 benchmark, gate fix now on main); one test conflict resolved, branch is MERGEABLE.

NGSIM real-data validation (Paper 1 §6.5 scaffolding)

Adds a real-traffic-data (NGSIM I-80) inverse-reconstruction path to attack the synthetic-noise caveat of the results above. Built TDD across 10 tasks (spec: docs/superpowers/specs/2026-07-13-real-traffic-data-ngsim-design.md, plan: docs/superpowers/plans/2026-07-13-real-traffic-data-ngsim.md); CI-inert (synthetic fixtures only — no raw NGSIM needed to build/test):

• applications/pinn/data/ngsim.py — Edie aggregation (trajectories → dense ρ(x,t)), objective monotone-window scan (reuses admissibility.violation, reports monotonicity_defect), Greenshields FD calibration, CLI → derived .npz (Git LFS; raw gitignored).
• core/problems/traffic_real.py — ngsim_wave Problem loading the .npz (lazy scipy; ground_truth = bilinear interp of the Edie field; smooth Greenshields f' for the residual).
• core/sampling.py — virtual loop-detector observation mode + disjoint held-out detectors.
• run.py/config.py/headline.py — --observations detectors + held-out-detector RMSE metric.
• baselines.py — Adaptive Smoothing Method (Treiber–Helbing) TSE baseline.
• figures.py --real — reconstruction slice + window heatmap + metric bars, computing ASM + RBF inline (PDF+PNG).
• RUNBOOK.md — reproduction commands, exploratory-gate note, ASM-tuning caveat.

Metrics mirror the synthetic sections (L¹/L²/admissibility) plus held-out-detector RMSE; monotonicity holds by disclosed window selection (single stop-and-go wave), with the reference's residual defect reported.

Remaining (human-in-the-loop, needs the raw NGSIM file): drop raw I-80 → preprocess → pass the exploratory gate → tune ASM → run headline + figures → write §6.5. The ASM baseline parameterization is deliberately deferred to that gate (documented in the RUNBOOK).

Fresh draft superseding the closed #98 (reopen was blocked after the main LFS history rewrite; branch content unchanged).