Recovering a scientific model's parameters from observations is an inverse problem. A useful implementation needs a parameterization, a residual definition, a sensitivity calculation, and a solver whose assumptions fit the objective.
A transparent calibration problem
Our worked example recovers amplitude, decay rate, and offset in an exponential-decay model. It uses noiseless synthetic observations with known parameters, making successful recovery independently checkable. Log parameters keep amplitude and rate positive; the offset remains unconstrained.
The notebook first compares the residual Jacobian with an analytic expression. That check is particularly useful for finding missing chain-rule factors caused by transformed parameters. It then performs functional Gauss–Newton steps and evaluates predictions at additional times.
Use curvature with the right contract
Gauss–Newton uses curvature formed from the residual Jacobian. It is generally different from the full Hessian of the nonlinear least-squares loss. Damping, initialization, residual scaling, and the linear solver all affect the behaviour of a practical implementation.
The omnibias optimizer accepts a residual closure and a flat parameter vector, returning a new vector and diagnostics. It does not silently update an arbitrary model object. Making this interface explicit keeps the numerical method and model state easy to inspect.
Beyond noiseless recovery
The example recovers its known parameters to floating-point precision, but experimental data presents additional questions: noise, correlated parameters, identifiability, outliers, parameter bounds, and model mismatch. Accurate synthetic recovery alone does not establish robustness or provide uncertainty estimates.
Working with Derivon
Derivon's scientific ML expertise can support calibration workflows from residual formulation and parameter constraints through implementation and validation. We help identify which questions a calculation actually answers and what additional evidence is needed before it informs an application. The open notebook provides a concrete starting point for that discussion.
Reproduce the result
The recorded notebook reports held-out rmse of < 1e-12. Inspect the conditions and raw data before comparing it with another method.
Read and run the complete notebook · Discuss a scientific ML project