No symbolic determining equations needed: sampled jet data recover an 11-dimensional intrinsic symmetry algebra
Synopsis
Continuous symmetries are usually found by writing the differential equations symbolically and solving an overdetermined determining system; here the search runs directly on sampled jet data through a linear operator that separates admissible generators from redundant directions. On a two-dimensional wave-equation benchmark the procedure recovers eleven generators in sequence, keeps validation symmetry residuals near machine precision, and reproduces Lie-bracket closure in a post-recovery check.
Interpretation
Symmetry generators can be recovered from jet-space data sampled on the equation manifold of a differential equation, without first giving a closed-form symbolic representation of the governing equations. The classical route prolongs a candidate generator, restricts it to the equation manifold, and splits the result with respect to unconstrained jet-fiber coordinates, yielding an overdetermined linear PDE system for the infinitesimal coefficients, the determining system. On the two-dimensional spatial wave equation, the sampled determining operator has a stated rank in the coefficient representation; after removing the additive solution sector the remaining represented dimension matches the known intrinsic symmetry dimension, and eleven generators are recovered sequentially.
The discovery task is recast as an operator problem: a finite admissible space is built first, and optimization runs only inside it rather than asking one optimizer to satisfy several objectives at once. Earlier formulations imposed symmetry, nontriviality, solution independence and novelty together as soft objectives; weakly constrained directions of the sampled residual Jacobian were hard to suppress, and residual errors of accepted generators propagated to later candidates. A fixed spline dictionary stays linear in its trainable coefficients, so the sampled determining operator becomes a fixed matrix whose nullspace does not move during training; on the wave-equation benchmark the eleven generators keep validation symmetry RMS near machine precision, several orders of magnitude below the acceptance threshold.
The span of the recovered generators is numerically closed under Lie brackets within the sampled domain, not merely eleven separately admissible directions. Earlier joint-learning approaches imposed bracket closure during discovery; here closure is checked only after recovery, so that legitimate directions are not excluded prematurely during the sequential procedure. For every unordered generator pair, one global constant-coefficient fit is evaluated on a disjoint off-shell validation set, and both the median and the maximum held-out relative closure error stay at a very small magnitude.
Perspective
The demonstrated setting is the linear two-dimensional spatial wave equation: samples are drawn on the equation shell rather than from one solution graph, the dictionary uses cubic B-splines with maximum interaction order 2, and the rank-stabilization preflight passes over three grid refinements. The authors frame the recovered algebra as a numerical approximation to the local intrinsic symmetry algebra within the represented and sampled domain, not as the full global symmetry group.
The authors state that the rank-stabilization preflight is an operational test at the representation level, and that stabilization does not prove the exact determining system has no unresolved functional freedom; the dictionary also has no established universal-approximation or density theorem, so a missing symmetry direction may reflect finite dictionary parameters rather than the equation. Extrapolation accuracy of the finite spline representation degrades away from the training support, which motivates patchwise transport as future work.