Эксперимент: Piola-Conditioned Fixed-Reference Neural Operator (#1159)

{ "worked": true, "confidence": 8, "verdict": "Built a fixed-reference surface-chart MVP with contravariant Piola transport, inverse transport, geometry conditioning, and a toy learned operator comparison. The numerical check reproduced exact area-weighted inner-product cancellation (relative error 0) and the discrete Piola divergence identity (maximum error 0) for valid shapes. Across two seeds, Piola conditioning reduced held-out relative physical-field error from 0.649 to 0.031 and from 0.627 to 0.036, respectively, a clear signal in this controlled toy setting. This is not evidence of a general Maxwell/operator win because the synthetic target was deliberately defined in reference coordinates.", "metrics": { "baseline": "Direct physical-component conditioning: relative test error 0.6486 +/- 0.0949 (seed 2973), 0.6274 +/- 0.0934 (seed 7311).", "idea": "Piola fixed-reference conditioning: relative test error 0.0313 +/- 0.0099 (seed 2973), 0.0364 +/- 0.0082 (seed 7311); math-check J range 0.6163 to 1.6294." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py", "files": [ "experiment.py", "experiment_seed2.py", "results.json", "article.md", "idea_context.json" ], "limitations": "Only a smooth square-chart synthetic operator was tested, not Helmholtz/Maxwell fields, remeshing, graph/Fourier neural operators, throughput, memory, batching size, stronger deformations, or invalid-shape rejection. The baseline uses raw physical components while the target is a fixed-reference operator, so the accuracy gap demonstrates representation alignment rather than a fair end-to-end PDE benchmark." }