Twindex

Your inverse solver returned a parameter. Twindex proves whether the data could have determined it.

Least squares, Bayesian inference and physics-informed neural networks always return a parameter. On their own, they do not give an independently checkable proof of whether a very different parameter fits your measurements just as well. Twindex finds such twins for the boundary value problem

-u''(x) + q(x; theta) f(u(x)) = g(x)  on (0, 1),   u(0) = alpha,  u(1) = beta,
q(x; theta) = sum_j theta_j B_j(x)   (Bernstein basis, theta >= 0),
f(u) = sum of a_m u^m over odd m, a_m >= 0,

and proves it: it writes a JSON certificate that a small exact checker (standard-library Python, no floating point, type-checked with mypy --strict) replays and verifies. A VALID pair certificate proves that no method can recover the parameter from those data with worst-case error below half the certified gap.

$ pip install twindex              # exact checker + CLI, zero dependencies
$ pip install "twindex[search]"    # + numerical search, audits and fitting (numpy)

$ twindex example theta500_pair > pair.json
$ twindex check pair.json
pair.json: VALID: ... every estimator of theta_0 has worst-case error >= 500 on these data

$ twindex audit --data measurements.csv --epsilon 1/1000 --prior 0,1000 --target 1 --output audit/

• Certificates: witness, pair (twins), box, and paving (two-sided bounds).
• Your model: any coefficient degree, polynomial sources, odd-power nonlinearities, inhomogeneous boundary values.
• twindex.fit: check any callable (a PINN, scipy, FEM) as an exact candidate.
• Every step is proved: https://arnav-m10.github.io/Twindex/THEORY/

Documentation: https://arnav-m10.github.io/Twindex/ · Source: https://github.com/Arnav-M10/Twindex