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NestyNet. IV. Laws Chosen by Nothing in Advance

NestyNet-DE is a novel framework that discovers complex, nonlinear differential equations and their underlying geometric symmetries directly from data without relying on fixed libraries, successfully recovering fundamental physical laws such as the reduced-Kepler hierarchy and Schrödinger equation through a combination of sparse-library and operator-factorized search strategies.

Original authors: Rodrigo Ibata, Wassim Tenachi, Foivos Diakogiannis, Neil Ibata, Anirudh Shankar

Published 2026-08-25

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CC BY 4.0

Original authors: Rodrigo Ibata, Wassim Tenachi, Foivos Diakogiannis, Neil Ibata, Anirudh Shankar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Science has long relied on a simple, powerful idea: that the universe speaks in the language of mathematics, and that if we can listen closely enough to the motion of things, we can hear the rules that govern them. For centuries, physicists have written down these rules as equations, describing how planets orbit, how light bends, and how particles interact. But finding these equations directly from raw data, without already knowing what to look for, has been one of the hardest challenges in modern computing. The difficulty lies in the fact that nature is rarely simple. Real-world systems often involve many moving parts that influence each other in complex, non-linear ways, and the data we collect is often noisy or incomplete. Traditional computer methods struggle here because they usually require a pre-made list of possible equations to choose from, or they rely on rough approximations of how things change, which can lead to false conclusions.

A new approach, detailed in a recent study by Rodrigo Ibata and his colleagues, offers a way to bypass these limitations. The researchers have developed a system that can listen to the motion of objects and reconstruct the exact mathematical laws that drive them, even when those laws are complex, involve multiple fields, or deal with quantities that have both real and imaginary parts. The key to their success is a method that treats the data not as a static list of numbers, but as a smooth, continuous story. By training a neural network to act as a perfect, noise-free version of the observed data, the system can calculate exactly how fast things are changing at any moment. This precision allows it to test potential laws with a level of accuracy that was previously impossible, separating the true governing rules from mere mathematical coincidences.

The researchers tested their system, which they call NestyNet-DE, on a wide variety of challenges, ranging from simple swinging pendulums to the complex equations that describe light and electricity. In one set of tests, they fed the system data from thirty years of daily observations of three hundred and eight asteroids orbiting in the main belt between Mars and Jupiter. These asteroids are not moving in a perfect, empty vacuum; they are constantly being nudged by the gravity of other planets, creating a messy, weathered record of their paths. Most computer models would struggle to find the underlying order in such a chaotic dataset. However, the new system successfully stripped away the noise and recovered the fundamental hierarchy of Kepler's laws. It identified the rule that a planet sweeps out equal areas in equal times, discovered the inverse-square law of gravity that pulls the asteroids toward the sun, and even reconstructed the energy equation that governs their motion. Remarkably, the system did not just fit numbers to a curve; it deduced a specific geometric symmetry in the data that forced the centrifugal term of the equation to take on a precise value, rather than just guessing it.

The study also demonstrated that the system could handle equations that describe the quantum world, where quantities are complex numbers, and the equations for electromagnetism, which involve vector fields like electric and magnetic forces. In these cases, the system had to find laws that were coupled together, meaning the behavior of one field directly dictated the behavior of another. The researchers showed that their method could recover the famous equations of Maxwell, which describe how light and electricity propagate, even when the data was noisy and the potential equations included many plausible but incorrect "decoy" terms. The system correctly identified the true terms and rejected the impostors, a feat that older methods failed to achieve at similar levels of noise.

What makes this work particularly significant is that it does not rely on a fixed dictionary of possible equations. Instead, it can invent new structural forms if the data demands them, searching directly over the architecture of the laws rather than just picking from a pre-written menu. It also goes beyond simply finding the equation; it can discover the hidden symmetries within that equation, such as the rotational symmetry that explains why angular momentum is conserved. By finding these symmetries directly from the data, the system can integrate the laws to find closed-form solutions and explain why the universe behaves the way it does. The researchers have made their code and data available to the public, allowing others to verify these results and apply the method to their own scientific questions. This represents a shift from simply fitting data to understanding the deep, structural logic of the physical world, offering a tool that can help scientists discover laws that no one had anticipated.

Technical Summary: NestyNet. IV. Laws Chosen by Nothing in Advance

Problem Statement
The discovery of governing differential equations (DEs) directly from data faces significant hurdles when moving beyond simple scalar systems to coupled fields, complex amplitudes, and structured Hamiltonian dynamics. Existing methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and PDE-FIND, rely on fixed libraries of candidate terms linear in unknown coefficients. This approach necessitates committing to specific functional forms (e.g., frequencies in sinusoidal terms) in advance and fails when the true law involves nonlinear parameters or compositional structures not present in the pre-defined dictionary. Furthermore, the accuracy of DE discovery is often bottlenecked by the quality of derivative estimation; standard numerical differentiation of noisy data or first-order trained neural surrogates introduces errors that can lead to physically incorrect laws. The challenge is to recover predictive, interpretable laws for coupled vector and complex-valued systems without being shackled to a fixed library or relying on numerically unstable derivatives.

Methodology
The paper introduces NestyNet-DE, a framework that integrates high-precision neural surrogates with two complementary discovery strategies and a symmetry-based feedback loop.

Analytic Derivatives via NestyNet Surrogates:
The foundation of the method is a trained neural surrogate (NestyNet) that fits trajectory or field data. Unlike standard multilayer perceptrons (MLPs) trained with first-order optimizers, NestyNet employs analytic derivatives (chain rule) through an Abstract Syntax Tree (AST) representation. This yields median accuracy improvements of $10^2$ – $10^3$ for derivatives compared to Adam-trained MLPs, ensuring that the feature matrix used for regression is derived from consistent, high-fidelity analytical gradients rather than noisy finite differences.
Two-Tiered Discovery Engine:
- Tier 1: Sparse-Library Route (STLSQ): This tier extends the SINDy/PDE-FIND paradigm. It constructs a feature matrix from an AST-based library of candidate terms (including powers, trigonometric functions, and vector operators) evaluated analytically on the surrogate. It employs Sequential Thresholded Least Squares (STLSQ) to select a sparse set of active terms. This tier handles coupled systems, complex-valued laws (via real decomposition), and vector equations with coefficient tying across components.
- Tier 2: Operator-Factorized Search (OFS): To address laws that no fixed library column can represent, this tier searches directly over equation structure. It treats the law as a balance of factorized terms $L[u] + \sum g_j(z_j)\phi_j = 0$ , where $\phi_j$ are fixed carriers (e.g., $\nabla \cdot$ , $\nabla \times$ ) and $g_j$ are discovered univariate functions. This route uses an archive-guided search to mutate and refine structural skeletons, allowing for the recovery of compositional laws (e.g., half-order kinetics, singular inverse-coordinate terms) that the sparse library misses.
Symmetry and Geometry Integration:
The framework discovers the Lie point symmetries of the recovered equation directly from the data. These symmetries serve three functions:
- Arbitration: They distinguish between candidates that forward rollout cannot separate.
- Library Expansion: They generate "compositional atoms" (e.g., specific coordinate-dependent coefficients) that are added to the library for re-selection.
- Integration: For solvable algebras, the discovered symmetries allow the integration of the recovered law into closed form. In phase space, Noether symmetries are used to derive conserved charges and structural coefficients (e.g., fixing the centrifugal term via rotational symmetry) rather than fitting them.
Refinement and Constraints:
The pipeline incorporates Variable Projection (VarPro) to optimize nonlinear shape parameters (e.g., exponents, rates) while analytically eliminating linear coefficients. Dimensional analysis is enforced as a pre-regression filter ("Buckingham–Sudoku"), rejecting terms that violate unit consistency, particularly crucial for coupled and vector systems.

Key Results
The framework was evaluated on a benchmark suite of 83 cases, including 57 real-valued ODEs and 26 complex-valued/coupled systems (e.g., Schrödinger, Dirac, Gross–Pitaevskii, and Maxwell's equations).

Scalar Benchmarks: The sparse-library tier alone passed 39 of 57 scalar cases. The addition of the operator-factorized search (Tier 2) recovered 13 additional cases involving complex nonlinearities and singularities, achieving a total pass rate of 52/57.
Complex and Vector Systems: The method successfully recovered the structure of complex-valued equations (handling modulus nonlinearities and coupled real/imaginary components) and vector PDEs. In the Maxwell equations case study, the framework correctly identified curl operators and rejected Laplacian "decoys" (false positives) up to $10^{-6}$ relative noise when using the NestyNet surrogate, whereas standard finite-difference methods failed at all noise levels.
Real-World Application (Asteroid Ephemerides): Applied to 30 years of daily ephemerides for 308 main-belt asteroids, the pipeline recovered the reduced-Kepler hierarchy. It identified the areal law, the inverse-square force, and the reduced Hamiltonian. Crucially, the discovered rotational symmetry allowed the derivation of the centrifugal coefficient ( $k = \ell^2$ ) from first principles rather than fitting it, demonstrating the method's ability to extract fundamental physical structure from perturbed, real-world data.

Significance and Claims
The authors claim that NestyNet-DE represents a significant step forward in scientific machine learning by:

Breaking the Fixed-Library Constraint: It recovers laws with nonlinear parameters and compositional structures that no pre-defined dictionary could anticipate, moving beyond the limitations of SINDy and PDE-FIND.
Handling Complexity: It is the first framework, to the authors' knowledge, to recover governing equations across a combination of real, complex-valued, and coupled vector laws directly from data while keeping the result explicit and interpretable.
Closing the Loop: The framework does not stop at the equation form; it recovers the geometry (Lie symmetries) of the law from the same data. This geometry is used to validate candidates, generate missing library terms, and structurally derive coefficients, effectively integrating the law.
Robustness to Noise and Derivatives: By relying on analytic derivatives from high-accuracy surrogates and enforcing dimensional consistency, the method achieves operational discovery accuracy in regimes where numerical differentiation fails.

The paper concludes that this approach provides a practical engine for interpretable equation discovery in physics, capable of handling the coupled, multi-field, and nonlinear dynamics that define much of modern theoretical physics.

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