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Discovering a governing equation

Because the activation derivative jet is exact, you can read y, y', y'', … directly off a fitted field and search for the implicit relation that generated your data — without choosing a named basis library up front.

Prerequisites​

pip install omnibias-symbolic

The idea​

Classical SINDy regresses derivatives against a fixed library of candidate terms. omnibias instead reads the derivatives exactly from the jet and searches for the relation directly — "library-free."

Step 1 — recover an ODE​

from omnibias.symbolic import discover_ode_law

# Suppose the data was generated by y' = 1 - y^2 (the logistic-tanh ODE).
result = discover_ode_law(t, y, max_order=2)
print(result["equation"])    # -> "y' = 1 - y^2"
print(result["coefficients"])

Step 2 — recover a PDE coefficient​

For field data, recover the PDE that links mixed partials. Here, the heat equation u_t = κ·u_xx:

from omnibias.symbolic import discover_field_pde_law, make_heat_field_split

train, val, test, hidden = make_heat_field_split(seed=0)
result = discover_field_pde_law(train, val, test, lhs_index=(0, 1), time_axis=1)
print(result["equation"])    # -> "u_t = 0.12*u_xx"

Step 3 — quantify confidence​

The discovery routines return more than an equation: coefficient uncertainty, model-order selection scores, and (optionally) certified intervals from the rigorous register.

print(result["coefficient_intervals"])   # certified bounds, when requested
print(result["selection_score"])
Discovery is a search, not an oracle

Recovered equations are hypotheses ranked by fit and parsimony. Validate on held-out data and inspect the coefficient intervals before trusting a discovered law. The closed-form jet removes derivative-estimation error from the pipeline — it does not remove modeling judgement.

What else is in the toolkit​

  • Multivariate vector-calculus and PDE-coefficient recovery (heat, wave, Burgers).
  • Dimensional analysis (Buckingham-Pi) and latent-state ODE discovery.
  • AutoML surrogates and information-geometry tooling.

These power the seven-chapter Discovery & Calculus Handbook in the source repo.

Next​

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