Field2Form: A Geometry-First Framework for Symbolic Solutions of Differential Equations, with a Held-Out Residual Court Against Extrapolation Blindness

Non-linear and complex differential equations govern physical phenomena across fluid dynamics, quantum mechan-ics, and astrophysics. While Physics-Informed Neural Networks (PINNs) and conventional Symbolic Regression (SR)algorithms excel at fitting observational data within training domains, they systematically suffer from “extrapolationblindness”—failing catastrophically when deployed beyond training boundaries or near boundary tails. In this work,we present Field2Form, an autonomous geometry-first framework that eliminates extrapolation blindness by couplingangle-field direction geometry, anchored latent neural generation, and a novel Held-out Tail Residual Test governed byKarush-Kuhn-Tucker (KKT) convex sparsity optimization. We establish two distinct evaluation protocols: Protocol A(Selection Gate Validation), which verifies the court’s capacity to isolate true analytical closed forms when present in can-didate libraries, and Protocol B (Unbiased Symbolic Discovery), where the framework autonomously discovers symbolicexpressions for forced, non-linear, and second-order differential systems without human-crafted dictionaries. Empiricalevaluations on the Logistic equation (y′= 1.1y(1 − y)), Riccati non-linear ODE (y′= y2− 1), forced non-linear ODE(y′= 1.1y(1 − y) +0.2 sin(x)), and the second-order non-linear pendulum (y′′+ sin(y) = 0) demonstrate that Field2Formachieves near-zero tail residual errors (MSE ≤ 10−23for analytical forms and MSE ≈ 10−4for dynamic harmonic series),while completely exposing and rejecting spurious polynomial or rational approximations whose tail errors explode byup to 5–6 orders of magnitude. Field2Form provides an end-to-end, reproducible, and fully autonomous pipeline forverifiable scientific discovery in physical systems.