40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 10: Inversion & Parameter Estimation (Simulation Stack + Inference)

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Chapter 10: Inversion & Parameter Estimation (Simulation Stack + Inference)

I. Objectives & Scope
This chapter unifies the forward simulation stack with statistical inference to estimate material/tension parameters θ via Bayesian/frequentist methods, perform model comparison, and deliver posterior diagnostics. It closes the loop with Chapters 6–8: critical/coherence windows, arrival-time data contracts, and the measurement matrix y = M(θ). All formulas/symbols/definitions are in English with backticks.

II. Forward Model & Observation Model (Unified Notation)

• S100-1 (Observation equation)
  y = M(θ, ν; x) + ε, where y is the observable vector (Chapter 8), θ are target parameters (lambda_L, xi, kappa_i, κ_ij, sigma1(ω), sigma2(ω), d, H_c1, H_c2, ...), ν are instrument/environment nuisances (modeled in cal_bundle), x are design variables (band/geometry/temperature/field/orientation/strain), and ε is noise.
• S100-2 (Path/arrival-time channels)
  For phase/group-delay channels, implement T_arr using exactly one of Chapter 7’s two conventions and record delta_form, gamma(ell), and d ell in data/report layers.
• S100-3 (Multi-channel stacking)
  y_stack = blkstack{ y^{(band, geom)} }, combined within coherence windows using weights w(ω) ∝ L_coh(ω) (Chapter 6).
• S100-4 (Tension hyperparameters)
  For thin-film/layered or nonlocal cases, introduce kernel-scale/strength hyperparameters η = {K_T, K_G, ...} entering via T_fil = T_loc + K * ζ into M(·) (Chapters 3, 4, 6).

III. Likelihood & Noise Models (Pluggable)

• S100-5 (Gaussian likelihood; block structure)
  ε ~ N(0, Σ_y), with Σ_y = blkdiag{ Σ_y(ω_k, geom_m)/w_k }; include piecewise terms from de-embedding/unwrap.
• S100-6 (Robust likelihood)
  For outliers/phase jumps use Student-t or Gaussian mixtures: p(ε) = ∑_c π_c N(0, Σ_c).
• S100-7 (Joint channels & calibration nuisances)
  Fold calibration nuisances into ν and update effective covariance Σ_y ← Σ_y + B Σ_ν B^T (B = ∂y/∂ν from Chapter 8).
• M10-1 (Likelihood calibration workflow)
  Use blanks/substrate-only references and repeats to estimate Σ_y, outlier rates, and coherence weights; output Σ_y, w(ω), outlier_mask.

IV. Priors & Physical Constraints

• S100-8 (Structural priors)
  kappa_i ≥ 1/√2, lambda_L > 0, xi > 0, sigma1(ω) ≥ 0; smoothness prior for sigma2(ω) (e.g., GP σ2 ~ GP(0, K(ω,ω'))).
• S100-9 (Layered/thin-film priors)
  d ~ N(μ_d, σ_d^2); kernel scales K_T, K_G as Half-Normal/Log-Normal.
• S100-10 (Hierarchical priors)
  Share hyperparameters across samples/batches: θ_i | ϕ ~ P(θ|ϕ), ϕ ~ P(ϕ).
• S100-11 (Design-tension coupling constraints)
  Sign/magnitude windows for ∂T_fil/∂ε, ∂T_fil/∂p (Chapter 9 S90-*), consistent with testable predictions.
• M10-2 (Prior integration)
  Encode physics/engineering priors in the dataset card under priors: with dimension checks.

V. Inference Algorithms & Compute Paths

• S100-12 (Targets)
  Posterior p(θ, η | y) ∝ p(y | θ, η) p(θ, η); evidence Z = p(y) = ∫ p(y | θ, η) p(θ, η) dθ dη.
• S100-13 (HMC/NUTS)
  For moderate dimension and differentiable M(·), use HMC/NUTS with mass-matrix/step-size adaptation; warm-up aligned to the Fisher structure.
• S100-14 (SMC/annealing/nested)
  For multi-modal/strongly nonlinear posteriors or evidence evaluation (thermodynamic integration / stepping-stone / nested sampling).
• S100-15 (Variational inference)
  ADVI / flow-based VI (normalizing flows) for rapid approximation and design iteration; calibrate with PSIS-LOO/Rank-SBC.
• M10-3 (NUTS pipeline)
  Build logp = log p(y|θ,η) + log p(θ,η) → gradients → preconditioning (via Chapter 8’s F) → adaptation → sampling/diagnostics.
• M10-4 (VI pipeline)
  Choose base family & transforms → maximize ELBO → sample approximate posterior → validate with PPC/LOO.
• I10-1 run_hmc(model, data, priors, init, adapt) -> {posterior, rhat, ess, traces}
• I10-2 run_vi(model, data, priors, family) -> {q(θ), elbo_hist, diag}
• I10-3 run_smc(model, data, schedule) -> {weighted_samples, logZ}

VI. Model Comparison & Evidence

• S100-16 (Candidate families)
  𝓜0: GL–London (local); 𝓜1: GL–London + tension-gradient term (λ_2); 𝓜2: nonlocal kernels {K_T, K_G} enabled; 𝓜3: hierarchical/batch-shared hyperparameters.
• S100-17 (Evidence / information criteria)
  Compute Z exactly/approximately (TI/SS/NS); use WAIC/PSIS-LOO/AIC/BIC for fast ranking.
• M10-5 (Comparison workflow)
  Estimate {logZ_k} and LOO_k for {𝓜k}; report Bayes factors/ranks with uncertainties.
• I10-4 compute_evidence(traces_or_q, method) -> {logZ, se}

VII. Posterior Diagnostics & Coverage

• S100-18 (Convergence & mixing)
  R̂ < 1.01, ESS/N > 50, stable energy/step statistics.
• S100-19 (PPC)
  Draw y_rep ~ p(y|θ); test whiteness/spectral slope/coherence-window consistency of residuals r = y − E[y|θ].
• S100-20 (SBC/coverage)
  Perform Simulation-Based Calibration on controllable subsets; nominal interval coverage near target levels.
• M10-6 (Diagnostic report)
  Auto-export convergence/correlation plots, PPC, and coverage; produce an “evidence–performance radar”.
• I10-5 posterior_predictive(traces_or_q, sim, N) -> {y_rep, metrics}

VIII. Simulation-Stack Integration (Forward Solvers & Interfaces)

1. S100-21 (Module set)
   • GL/London: steady/linearized solutions of S40-* (FEM/spectral);
   • EM propagation: cavity/resonator modes, waveguide/thin-film S21 (FDTD/TL/eigenmode expansion);
   • Vortices & TDGL: coarse-grained integration of S50-*;
   • Thin-film/nonlocal: kernel convolution & thickness scaling (K_T, K_G).
2. S100-22 (Auto-diff & Jacobians)
   Provide ∂y/∂θ, ∂logp/∂θ for HMC/VI; for black-box solvers use adjoints/finite-diff with sparse approximations.
3. M10-7 (Simulation–inference coupling)
   Map SimStack outputs to measurement-matrix channels under a unified API with dimension checks.
4. I10-6 simulate_forward(θ, η, x, channels) -> {y_hat, J, meta}

IX. Synthetic Data & Benchmarks

• S100-23 (Benchmark suite)
  Four representative systems (Cuprates / Fe-based / Heavy-Fermion / 2D) with known {θ_true, η_true} and multi-channel observations.
• M10-8 (SBC/benchmark regressions)
  Evaluate bias/variance of HMC/VI/SMC across noise/drift/outlier regimes; build a “method selection table”.
• I10-7 make_synthetic(case_id, noise, drift, outliers) -> {data, truth}

X. Outputs & Data Contract (Inference Section)

• S100-24 (Dataset-card fields)
• inference:
• model: ["GL-London","GL+gradT","Nonlocal","Hierarchical"]
• likelihood: {type: "Gaussian|StudentT|Mixture", Sigma: "...", weights: "..."}
• priors:
• lambda_L: "HalfNormal(μ,σ)"
• xi: "HalfNormal(μ,σ)"
• kappa_floor: "1/sqrt(2)"
• K_T: "LogNormal(μ,σ)"
• sampler: {algo: "NUTS|SMC|VI", steps: 2000, warmup: 1000, adapt: true}
• diagnostics: {rhat_max: 1.01, ess_min: 50}
• evidence: {method: "TI|SS|NS|WAIC|LOO"}
• contracts:
• T_arr: {convention: "pulled_const|integrand", delta_form: "...", gamma: "...", d_ell: "..."}
• references:
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.6"
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.7"
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.8"
• M10-9 (Reproducibility bundle)
  Package scripts/parameters/environment digest and anchor list; validate check_dim and citation style.

XI. Interfaces to Adjacent Chapters

• With Chs. 3–4: forward equations and tension couplings (P30-*, S40-*).
• With Ch. 6: outputs of W_c/W_coh and M6-* (derived quantities/confidence intervals).
• With Ch. 7: path/measure declarations for T_arr and de-embedding/unwrapping outputs.
• With Ch. 8: measurement matrix J, Fisher information, and design-optimization entry.
• With Ch. 11: case-library dataset cards for validation and falsification.

XII. Anchors (This Chapter)
S100-1—S100-7 (observation/likelihood); S100-8—S100-11 (priors); S100-12—S100-15 (algorithms); S100-16—S100-20 (comparison & diagnostics); S100-21—S100-24 (sim-stack & contracts).
M10-1—M10-9 (likelihood calibration, HMC/VI/SMC pipelines, comparison, diagnostics, sim coupling, SBC, reproducibility).
I10-1—I10-7 (run/evidence/predict/simulation & synthetic-data APIs).

XIII. Summary
Under a unified data contract and measurement matrix, this chapter couples forward simulation with statistical inference to deliver metrologically sound estimates of θ/η, comparable model evidence, and iteratively optimized experiment design. Together with Chapters 6–8 and the case library, Chapter 11 can directly reuse these outputs for cross-system validation and falsification.