TL;DR
This paper introduces a symmetry-informed pipeline for discovering governing PDEs from data by leveraging differential invariants, significantly reducing the search space and improving discovery accuracy.
Contribution
It proposes a novel method using differential invariants to guide PDE discovery, ensuring symmetry is strictly incorporated and enhancing success rates.
Findings
Outperforms existing symmetry-informed PDE discovery methods.
Reduces search space without loss of information.
Achieves higher success rate and accuracy in PDE identification.
Abstract
The explicit governing equation is one of the simplest and most intuitive forms for characterizing physical laws. However, directly discovering partial differential equations (PDEs) from data poses significant challenges, primarily in determining relevant terms from a vast search space. Symmetry, as a crucial prior knowledge in scientific fields, has been widely applied in tasks such as designing equivariant networks and guiding neural PDE solvers. In this paper, we propose a pipeline for governing equation discovery based on differential invariants, which can losslessly reduce the search space of existing equation discovery methods while strictly adhering to symmetry. Specifically, we compute the set of differential invariants corresponding to the infinitesimal generators of the symmetry group and select them as the relevant terms for equation discovery. Taking DI-SINDy (SINDy based on…
Peer Reviews
Decision·Submitted to ICLR 2026
This paper offers a theoretically sound and elegant contribution to the field of equation discovery. By leveraging differential invariants derived from known or learned symmetries, the authors present a clear and general framework that can be applied to improve many existing methods such as SINDy. The theoretical foundation, grounded in classical Lie symmetry analysis, is both rigorous and well-integrated into the machine learning context. The conceptual innovation of embedding symmetry directly
Despite its theoretical depth, the paper’s empirical scope is somewhat narrow. The experiments are limited to one-dimensional canonical PDEs such as Burgers, KdV, and KS, leaving open questions about scalability to higher-dimensional or real-world systems. The claim of computational efficiency is also not empirically validated, while the method does reduce the search space, the additional overhead of computing differential invariants is not benchmarked. A runtime or complexity analysis would hel
1. The algorithm systematically introduces the theory of differential invariants into data-driven equation discovery, demonstrating significant theoretical depth and methodological innovation. 2. The method is built upon a solid mathematical foundation, ensuring strict adherence to symmetry constraints. 3. It features a plug-and-play nature that allows for integration with existing discovery methods, offering strong practicality and exhibiting strong robustness against noise and symmetry deviati
1. The method's performance is highly dependent on the accuracy of the symmetry information. For complex systems where the symmetries are unknown or difficult to discover, the practical applicability of the method may be limited. 2. The discussion is confined to Lie point symmetries. Compared to a broader perspective on symmetry in machine learning (e.g., as explored in works like "Machine-learning hidden symmetries" by Ziming Liu and Max Tegmark, which deals with Translation invariance, Lie inv
A strong concept for model discovery if it holds up under more realistic conditions.
There seems to be no noise study on the method a the method itself really does have to be evaluated on this idea. More broadly, there are works that are pretty similar, even tough this is a new architecture, but perhaps for a conference like ICLR a higher level of distinction of novelty is required in comparison with other methods.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
MethodsSparse Evolutionary Training
