Entropy-Symmetrized Neural Flux MVP

What is implemented

experiment.py contains:

• A = solve(H, S) with S = S.T and H = L L.T + eps I;
• Gauss-Legendre path integration from the zero reference state;
• a direct MLP flux control;
• a small smooth two-state flux-fitting experiment;
• numerical checks of the similarity transform and antisymmetric contamination.

The implementation uses CUDA when available and retries on CPU if CUDA execution fails. The fixed seed is 1729.

Mathematical verification

For random symmetric S and positive-definite H, the experiment compares eigenvalues of A = H^{-1}S against those of B = H^{-1/2} S H^{-1/2}. Across condition numbers 1 through 1e12, the largest imaginary part of A was zero in double precision. The eigenvalue mismatch was approximately 1.6e-15 at condition 1 and 9.3e-6 at condition 1e12, with a corresponding similarity residual of 7.7e-6. This is consistent with the exact identity and increasing floating-point sensitivity under ill-conditioning.

As a control, adding a random antisymmetric component to S produced complex eigenvalues. At contamination scales 0.5, 1, 2, and 4, the measured complex fractions were 0.16, 0.50, 0.76, and 0.855, close to the analytic predictions 0.147, 0.469, 0.717, and 0.856.

Mini-experiment

On 900 uniformly sampled two-state inputs, both models were trained for 900 Adam steps and evaluated on 300 held-out inputs. The entropy-symmetrized model used 585 parameters versus 722 for the direct MLP.

| model | test flux MSE | max sampled Jacobian imaginary part | |---|---:|---:| | direct MLP | 9.2103e-5 | 0.0 | | entropy-symmetrized | 3.2301e-5 | 0.0 |

The constrained model therefore showed lower test flux error in this run, while both models had real sampled Jacobian spectra. The direct model's real spectrum is a property of this particular fitted toy target, not a general guarantee.

Honest conclusion

The claimed algebraic phenomenon is clearly reproduced: exact symmetry plus positive definiteness gives real characteristic speeds, and violating symmetry can yield complex speeds. There is a promising small flux-regression signal, but this MVP did not test a finite-volume solver, long-horizon rollouts, CFL violations, or Riemann problems; consequently it does not establish a practical stability win in a PDE simulation.

Run with:

/home/maxwelhelp/main/bin/python3 experiment.py