Inverse-PINN physics discovery

Identify the governing parameters of a nonlinear dynamical system from data with a physics-informed inverse engine — and get a self-doubt RED gate when a parameter is not identifiable.

Headline number: 1 POST → the Duffing cubic stiffness α recovered ≈ 0.989 (truth 1.0), GREEN, grad-norm 2e-16, with a signed receipt.

The Duffing oscillator (m,x'' + c,x' + \delta,x + \alpha,x^3 = F\cos(\omega t)) is the canonical nonlinear test system. Given a trajectory, the inverse problem is: recover (\alpha) (and friends). The SZL inverse-PINN does this with a NumPy-only spectral surrogate (Fourier modes + polynomial, exact analytic derivatives) — no torch, no DeepXDE — and gates the answer on identifiability.

Honest scope. Recovered values are MODELED (fit to data; not measured). The convergence gate (causal weight, gradient norm, FIM condition number, Fisher floor) is REAL. The F19 Bekenstein bound is a PROVEN inequality (locked-8 @ c7c0ba17); its application here is MODELED with SAMPLE R, E. Λ is Conjecture 1 (advisory, ≤ 0.99). Live a11oy Space.

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Prerequisites

python3 -m pip install httpx

Live base: https://a-11-oy.com. Probe the organ:

curl -s https://a-11-oy.com/api/a11oy/v1/pinn/health | jq '{ok, supported_systems, self_doubt_gate: .honesty.self_doubt_gate}'

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Quickstart (live, verified)

import httpx
BASE = "https://a-11-oy.com"

r = httpx.post(f"{BASE}/api/a11oy/v1/pinn/identify", json={"demo": "duffing"}, timeout=120).json()

print("system:", r["system"], "| convergence:", r["convergence"]["label"])   # duffing | GREEN
alpha = r["discovered"][0]
print(f"{alpha['name']} = {alpha['value']:.4f}  CI95 {alpha['ci95']}  label {alpha['label']}")
# => alpha = 0.9894  CI95 [0.942, 1.014]  label MODELED
print("grad_norm:", r["convergence"]["grad_norm"])        # ~2e-16 (GREEN < 1e-05)
print("kappa_fim:", r["convergence"]["kappa_fim"])         # ~1.0 (well-conditioned ⇒ identifiable)
print("lambda_advisory:", r["lambda_advisory"]["value"], r["lambda_advisory"]["status"])
print("Bekenstein:", alpha["bekenstein"]["label"])         # PHYSICALLY_PLAUSIBLE

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Full walkthrough

Step 1 — The surrogate

The engine fits a spectral surrogate to the trajectory (Fourier 24 modes + polynomial deg 3) so that derivatives are exact analytic rather than finite-differenced. Linear parameters solve by exact least-squares; nonlinear ones by Adam gradient descent on the physics residual.

Step 2 — The self-doubt gate

This is the governance heart. A fit is only asserted GREEN when all hold:

Criterion	GREEN threshold
min_causal_weight	> 0.99 (RED ≤ 0.5)
grad_norm	< 1e-05
kappa_fim	< 1e+06 (IDENT); ≥ 1e+08 ⇒ RED
min_fisher	above 1e-08 floor; below ⇒ UNIDENTIFIABLE

A non-identifiable parameter (Fisher below floor or FIM ill-conditioned) is labelled RED/UNIDENTIFIABLE and not asserted. The engine refuses to pretend it knows.

Step 3 — Read the discovery

Each parameter ships value, analytic ci95, std, fisher_information, identifiable, convergence_label, and a Bekenstein plausibility check. The Duffing α comes back ≈ 0.989 with a tight CI bracketing the ground-truth 1.0.

Step 4 — The receipt

receipt.payload is a szl.lake.receipt/v1 envelope (organ a11oy-pinn, kind inverse_pinn_identify) recording the system description, method, and discovered parameters, hash-chained into szl-lake. Cross-check the head:

print(httpx.get(f"{BASE}/api/lake/v1/health", timeout=30).json()["organs"]["a11oy-pinn"]["chain_head"])

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Honest scope table

Claim	Status
Parameter recovery (α, …)	MODELED — fit to data, not measured
Convergence / identifiability gate	REAL — causal weight, grad norm, Fisher, κ(FIM)
F19 Bekenstein bound	inequality PROVEN (locked-8); application MODELED
Λ advisory	Conjecture 1 — advisory, ≤ 0.99, never a proof
Receipt	SHA3-256 chained; signed or honest DSSE_PLACEHOLDER

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See also

• 16 — CALPHAD inverse-discovery — same engine, materials vertical.
• 18 — Governed materials prediction — novelty + certify.
• 12 — Doctrine ledger query — assert the locked numbers live.

Cite this recipe

@misc{szl_cookbook_inverse_pinn_2026,
  title        = {Inverse-PINN physics discovery (SZL Cookbook recipe 19)},
  author       = {{SZL Holdings}},
  year         = {2026},
  howpublished = {\url{https://github.com/szl-holdings/szl-cookbook/blob/main/recipes/19-inverse-pinn-physics-discovery.md}},
  note         = {Duffing parameter ID; values MODELED; self-doubt RED gate REAL. Λ = Conjecture 1.}
}

References: Raissi, Perdikaris & Karniadakis 2019, J. Comput. Phys. 378:686 (PINNs); Duffing 1918. Bekenstein 1981, Phys. Rev. D 23:287 (F19, locked-8 @ c7c0ba17).

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Doctrine v11 LOCKED — 749/14/163 — kernel c7c0ba17 · Λ = Conjecture 1 · SLSA L1 (honest)