fix: heat identification_error → area-weighted L2 integral (#85)#93
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andrinr wants to merge 3 commits into
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fix: heat identification_error → area-weighted L2 integral (#85)#93andrinr wants to merge 3 commits into
andrinr wants to merge 3 commits into
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Addresses #85 (WIP — needs the thermal-mesh benchmark to verify). identification_error was reported as a raw nodal sum sum((T-T*)²) in every heat solver's forward, while the adjoint-based solvers (Firedrake, FEniCS) differentiated the area-weighted integral ∫(T-T*)²dΩ and multiplied the result by an approximate n_nodes/domain_vol scalar to bridge the two. That scalar is only ~5% accurate, so the returned gradient was not the exact derivative of the reported objective; L-BFGS accumulates these inconsistent gradients into a corrupted curvature estimate and converges to a suboptimal point. Make ∫(T-T*)²dΩ the reported objective everywhere, so the gradient is the exact derivative of what is reported: - firedrake-heat, fenics-heat (adjoint): the forward now reports the same assemble(inner(T-T*,T-T*)*dx) the adjoint already differentiates, and the n_nodes/domain_vol correction factor is removed (both rho and source gradients). Internally exact. - jax-fem, torch-fem-thermal (autodiff): the forward objective is weighted by lumped nodal volumes derived from the shared hex mesh; since AD runs straight through the objective, the gradient is automatically consistent. - dealii-heat (forward-only): same lumped-volume weighting applied to the reported scalar. The lumped-volume helper splits each axis-aligned hex cell's bounding-box volume equally among its 8 nodes (verified: total weight = domain volume). Firedrake/FEniCS use the consistent mass matrix while the other three use lumped mass; these differ at O(h²) and both converge to the same continuous objective. Reconciling to one convention (and adding a deal.II adjoint) is left as follow-up. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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📊 View the full benchmark resultsThe rendered docs preview has every plot for this run (forward accuracy, Status diff vs baseLegend · ✅ ok · 🟠 anom · ❌ fail · · missing · 🚫 excluded (permanent — out of score denominator) · ⚪ excluded (work-to-do) · * stale — result predates current benchmark run No status changes. · score 0.98 → 0.99 Full Mosaic statusMosaic statusLegend · ✅ ok · 🟠 anom · ❌ fail · · missing · 🚫 excluded (permanent — out of score denominator) · ⚪ excluded (work-to-do) · * stale — result predates current benchmark run Each solver is run against every experiment in the suite. ok = produced valid results; fail = crashed or returned invalid data; anom = ran successfully but tripped an automated quality check (e.g. poor gradient accuracy, outlier wall-clock time, or diverged optimisation). Thresholds are defined per-problem in the problem config.
Failures & anomalies
ns-3d-grid — 16 experiment(s)
ns-grid — 20 experiment(s)
structural-mesh — 10 experiment(s)
thermal-mesh — 14 experiment(s)
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Root cause
identification_errorwas the raw nodal sumΣ(T-T*)²in every forward, but the adjoint solvers (Firedrake, FEniCS) differentiated the area-weighted integral∫(T-T*)²dΩand multiplied by an approximaten_nodes/domain_volscalar. Per the paper excerpt in the issue, that scalar is only ~5% accurate, so the gradient isn't the exact derivative of the reported objective — and L-BFGS folds the inconsistency into a corrupted Hessian estimate, converging to a suboptimal point.Fix: make
∫(T-T*)²dΩthe reported objective everywhereassemble(inner(T-T*,T-T*)*dx)(plumbed through_solve_heat); removednodal_correctionon both rho & source gradsjax.vjpthrough the objectivetorch.autogradThe shared
_lumped_node_weightshelper splits each axis-aligned hex cell's bounding-box volume equally among its 8 nodes. Unit-tested: total weight = domain volume; interior nodes (shared by 2 cells) weighted 2× corners.Known limitations (why it's a draft)
*dx); the other three use lumped mass. These agree to O(h²) and converge to the same continuum objective, but aren't bit-identical. Reconciling to one convention is follow-up.identification_error/final_ic_errornumbers in published results.