Title:NestyNet. IV. Laws Chosen by Nothing in Advance

Abstract:Differential-equation (DE) discovery tends to break down precisely where much of physics begins. Fields are coupled, governing laws are nonlinear in the state, amplitudes, coordinates, or operators of interest, yet derivatives must remain consistent across fields, channels, and differentiation orders. NestyNet-DE addresses this via two complementary DE search strategies, both employing analytic derivatives from segmented neural surrogates: a sparse-library route linear in the outer coefficients, and a new operator-factorized route that searches directly over equation structure rather than a fixed library, recovering compositional laws the sparse route misses. The recovered laws can be strongly nonlinear in states, fields, coordinates, and their couplings. The framework also handles multi-dataset shared-support discovery, complex and vector laws, and Hamiltonian discovery from phase-space trajectories. Beyond a law's form, the same data yield its geometry, the Lie point symmetries of the recovered equation. We apply the pipeline to 30 years of daily ephemerides of $308$ main-belt asteroids and recover the reduced-Kepler hierarchy (areal law, inverse-square force, and reduced Hamiltonian), where a discovered rotational symmetry fixes the centrifugal coefficient rather than fitting it. We also present a benchmark of 57 real-valued ODEs and 26 complex-valued systems, including the Schrödinger and Dirac equations, with Maxwell's equations as a coupled vector-PDE case study. Here, discovery is not shackled to a fixed library, nor does it end at the equation. It composes laws that no dictionary anticipated, and closes the loop from data, to a law chosen by nothing in advance, to the geometry that explains and integrates it.