Discovering Symbolic Differential Equations with Symmetry Invariants

TL;DR

This work tackles the challenge of discovering symbolic partial differential equations from data by imposing symmetry via differential invariants. By replacing the original variables with a complete set of invariants for a given symmetry group, the authors constrain the symbolic regression search space to symmetry-respecting models, and they demonstrate integration with sparse regression and genetic programming. The approach yields more accurate and parsimonious PDEs across fluid, Darcy flow, and reaction-diffusion systems, including robustness under noise and imperfect symmetry through relaxation. The framework advances interpretable, physics-consistent equation discovery with practical gains in efficiency and reliability for data-driven dynamical modeling.

Abstract

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of \textit{symmetry invariants} in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.

Figures (7)

• Figure 1: Our framework enforces symmetry in equation discovery by using symmetry invariants. We highlight three discovery algorithms in their original form (bottom row) and when constrained to only use symmetry invariants (top row). The colored circles visualize the predicted functions on a circular domain and demonstrate that using symmetry invariants guarantees a symmetric output.
• Figure 2: Venn diagram of hypothesis spaces from base SR methods and our symmetry principle.
• Figure 3: Success probabilities of sparse regression methods on the reaction-diffusion system with noisy data (left), unequal diffusivities (center) and external forcing (right). Under noisy data, our method (SI) consistently outperforms SINDy under the same number of test functions. For systems with imperfect symmetry, strictly enforcing symmetry (SI) can hurt performance, but a relaxed symmetry constraint (SI-relaxed) is still better than no inductive bias (SINDy).
• Figure 4: Basis for the SINDy parameter subspace that preserves $\mathrm{SO}(2)$ symmetry $\mathbf v = -v\partial_u + u\partial_v$. The SINDy parameter $W$ has dimension $2 \times 19$. The two rows correspond to the two equations with $u_t$ and $v_t$ as the LHSs. The RHS contains $19$ features, including all monomials of $u,v$ up to degree $3$ and their spatial derivatives up to order $2$. The set of symmetry invariants used to compute the basis is given by $\{ t,x,y,u^2+v^2 \} \bigcup \{ \mathbf u\cdot\mathbf u_\mu \}\bigcup \{\mathbf u^\perp \cdot \mathbf u_\mu\}$, where $\mathbf u=(u,v)^T$ and $\mu$ is a multiindex of $t,x,y$ with order no more than $2$. The top $7\times 2$ grid displays the original basis solved from SVD, and the bottom $7\times 2$ grid displays the sparsified basis.
• Figure 5: Success Probabilities of GP-based methods on different systems. Our method with symmetry invariants can discover the correct equations with fewer iterations.

Theorems & Definitions (15)

• Definition 2.1
• Definition 3.1: Def 2.51, olver1993applications
• Theorem 3.2: Prop 2.56, olver1993applications
• Proposition 3.3
• Proposition 3.4
• Theorem B.1: Thm. 5.48, olver1995equivalence
• Proposition B.2
• proof
• Remark B.3
• Remark B.4