FlatSeek hands flat-output discovery to a language model: 74 literature systems solved, 171 synthetic
Synopsis
Planning and control of differentially flat systems rests on a flat output, yet no general procedure writes one down for arbitrary nonlinear dynamics. FlatSeek reverses the reconstruction relation in the definition of flatness: fix an output and its reconstruction maps, close the dynamics differentially, and obtain system–output pairs with known certificates at scale. A Qwen3.5-2B model fine-tuned with LoRA proposes candidates that a symbolic verifier accepts or rejects. FlatSeek solves 74 literature and 171 synthetic systems, against 58 and 127 for the strongest traditional symbolic method.
Interpretation
Finding a flat output moves from manual trial and error per system to a fine-tuned language model proposing explicit symbolic candidates that a verifier screens. Flat outputs have no fixed form, and some systems necessarily require input derivatives of unknown order; static or dynamic feedback linearization, dynamic extension and pure input prolongation each apply only when their structural conditions hold, while necessary conditions and order bounds can screen candidates but not construct outputs. FlatSeek solves 74 systems on the literature benchmark against 58 for the strongest traditional symbolic method, and 171 against 127 on the synthetic benchmark, with every method judged by the same candidate-checking protocol and a system counted as solved when at least one candidate is accepted.
Training data are generated backward from the output side rather than labeled: an output and finite-order reconstruction maps are fixed first, and differential closure yields dynamics for which that output is flat. Labeling sampled systems forward is itself the discovery problem being solved, and no public corpus supplies system–output pairs, so supervision cannot come from sampling dynamics and annotating them. The base corpus holds accepted system–output pairs, and symbol renaming plus state permutation augment it into the supervised fine-tuning corpus; synthetic test systems are held out and structurally disjoint, sharing no structural fingerprint with any training system.
At inference the model only has to propose several candidates, and acceptance is decided solely by a symbolic verifier that differentiates each candidate along the dynamics and applies a finite-order Jacobian-rank test. Untuned reference models reach 50 to 67 systems on the literature benchmark, and Qwen3.5 gains little from 9B to 27B, so candidate quality is not recovered by model scale alone. The fine-tuned model is Qwen3.5-2B with LoRA, freezing base weights and updating only low-rank adapters; all language models receive the same prompt and their officially recommended decoding settings, with a fixed candidate budget per system.
Perspective
Evaluation covers a literature benchmark of published systems and a synthetic benchmark generated by backward construction and structurally disjoint from training, with all methods judged by the same verifier and undecidable candidates counted as failures. The authors propose extending the search to systems for which existing methods have not yet produced a flat-output candidate.
Synthetic flat outputs exist by construction, and models see only dynamics, not reference outputs; literature hits depend on verifier judgment, and candidates the verifier cannot decide count as failures. The authors list extension to systems where existing methods have no candidate as future work, so current coverage stays inside that boundary.