Many physical laws are simple only after the right representation, decomposition or internal coordinate has been found, but discovering that structure from data is combinatorially hard. This task is symbolic regression (SR), the search for closed-form expressions that fit data without assuming a fixed model class. Here we present NestyNet-SR. A neural surrogate with analytic derivatives is used to detect separability, recursively reducing multivariate problems to simpler neural atoms. These atoms are distilled into closed form by a tiered symbolic-search stack, whose final tier is a novel factorized symbolic search that separates structure from calibration. Composing candidate internal coordinates freely, it scores each coordinate by how well calibrated functions of it (e.g., polynomials, power laws, sinusoids) fit the data, so the constants of those calibrated maps, however deeply nested in the final expression, are fitted rather than searched. The method supports multi-dataset regression, automated feature discovery, and dimensional-analysis pruning. On the SRBench AI~Feynman benchmark, NestyNet-SR achieves exact symbolic recovery of all 120 noiseless equations, the first such result, and under noise a statistical audit certifies which structures survive. As a real-data vignette, given only the separate mass-model components of SPARC-survey galaxies, the algorithm discovers the baryonic acceleration coordinate, reproduces the established mass-to-light and acceleration scales and the non-unique form of the radial acceleration relation, and adds held-out-galaxy generalization, a calibrated symmetry abstention, and a posterior for the local slope of the law. Analytic derivatives thus provide a practical route from neural surrogates to interpretable closed-form empirical laws.
Rodrigo Ibata, Wassim Tenachi, F. Diakogiannis et al.· 0 citations
Many of the smooth functions that matter most in physics are precisely the ones that standard neural network methods struggle to fit accurately. Here we present NestyNet, a coupled model-and-optimizer framework capable of fitting such targets to high accuracy while also delivering their gradients, Hessians, Laplacians, and antiderivatives analytically and at low cost. This makes it a natural substrate for scientific machine learning tasks. The model is a deterministic segmented analytic surrogate, and its optimizer is a second-order Levenberg--Marquardt scheme whose damping and linear solves are tailored to the stiff, strongly correlated parameter geometries induced by multiscale and sharply structured targets typical in physics and other scientific applications. On the AI Feynman benchmark of 120 physics equations, NestyNet achieves median improvement factors of $2\,100\times$ for function values, $1\,400\times$ for first derivatives, and $780\times$ for second derivatives relative to standard neural networks trained with first-order optimization (Adam). Even after refining those fits with quasi-Newton (L-BFGS) optimization, the corresponding improvements are $540\times$, $450\times$, and $250\times$. Owing to the analytic design it is up to $\approx 44\times$ faster than vectorized automatic-differentiation (autograd) baselines, with a margin growing with model size. The same analytic-derivative framework also supports vector- and complex-valued targets, measurement uncertainties in both inputs and outputs, and constraints, and its modules can be composed flexibly to build scientifically useful model architectures, all without reverting to autograd. Together, these components provide a practical modular framework for fitting difficult scientific surrogates while delivering accurate differential operators for subsequent analysis.
Rodrigo Ibata, Wassim Tenachi, F. Diakogiannis et al.· 0 citations
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 Schr\"odinger 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.
Rodrigo Ibata, Wassim Tenachi, F. Diakogiannis et al.· 0 citations
A low-dimensional posterior for the fitted function itself, for scientific neural surrogates trained with second-order optimization, provides a highly efficient route to uncertainty propagation for derivative-dependent scientific inference.
Rodrigo Ibata, Wassim Tenachi, F. Diakogiannis et al.· 0 citations
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