NestyNet. IV. 无预先选定的定律
NestyNet. IV. Laws Chosen by Nothing in Advance
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中文总结 AI 辅助
NestyNet-DE 采用两种互补的微分方程搜索策略，通过解析导数实现方程结构直接搜索，应用于小行星星历恢复开普勒层级，完成从数据到定律再到几何的闭环。
中文摘要 AI 辅助
微分方程（DE）发现往往在物理学的起点处失效：各场相互耦合，控制定律对所关注的状态、振幅、坐标或算子呈非线性，且导数需在各场、各通道及各微分阶间保持一致。NestyNet-DE 采用两种互补的 DE 搜索策略解决该问题，均基于分段神经代理的解析导数：一是外层系数呈线性的稀疏库路径，二是新的算子分解路径，该路径直接搜索方程结构而非固定库，可恢复稀疏路径遗漏的组合定律，且恢复的定律对状态、场、坐标及其耦合可呈强非线性。该框架还可处理多数据集共享支持发现、复杂与矢量定律，以及相空间轨迹的哈密顿量发现。除定律形式外，相同数据还可得出其几何结构，即恢复方程的李点对称性。我们将该流程应用于 30 年间 308 颗主带小行星的每日星历，恢复出简化开普勒层级（面积定律、平方反比力、简化哈密顿量），其中发现的旋转对称性确定了离心系数而非拟合得到。我们还给出了 57 个实值常微分方程（ODE）和 26 个复值系统的基准测试，包括薛定谔方程、狄拉克方程，并以麦克斯韦方程作为耦合矢量偏微分方程（PDE）的案例研究。在此，发现不受固定库的束缚，也不局限于方程本身，它可生成字典未预见的定律，完成从数据到无预先选定的定律、再到解释并整合该定律的几何结构的闭环。
英文摘要
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.
