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Explicit Discovery of Nonlinear Symmetries from Dynamic Data
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Proposes LieNLSD, which discovers the explicit Lie algebra subspace (dimension and nonlinear infinitesimal generators) of PDE/dynamic-data symmetries by solving a linear system via SVD on a neural network's Jacobian and finite-difference derivatives.
Problem
Discovering nonlinear symmetries from dynamic or static data, including the dimension of the Lie algebra subspace and explicit forms of infinitesimal generators, which prior methods either restrict to linear symmetries or do not express explicitly.
Approach
Specify a function library for infinitesimal generators, prove the prolonged generator stays linear in the coefficient matrix W, then solve the linear infinitesimal criterion (using central-difference derivatives and the Jacobian of a trained MLP) by SVD; a LADMAP step sparsifies the basis.
Key results
- Grassmann distance to ground-truth linear symmetry subspace for LieNLSD is lower than LieGAN on all four datasets, e.g., 0.092 vs 0.251 on top quark tagging and 0.0126 vs 1.58 on Burgers' equation.
- FNO with LPSDA using LieNLSD symmetries improves long rollout NMSE over no augmentation by over 20% on Burgers', heat, and KdV (e.g., heat 0.107 to 0.0599).
- LieNLSD recovers 7 generators for top quark tagging (Lorentz plus scaling), 6 for Burgers', 20 for the 2D wave equation, and 6 for Schrödinger, matching known symmetry counts.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LieNLSD instead of LieGAN for symmetry discovery | Symmetry discovery · Top quark tagging | Grassmann distance: (2.51 \pm 0.41) \times 10^{-1} -> (9.20 \pm 1.83) \times 10^{-2} | Helped |
| Use LieNLSD instead of LieGAN for symmetry discovery | Symmetry discovery · Burgers' equation | Grassmann distance: 1.58 \pm 0.05 -> (1.26 \pm 0.20) \times 10^{-2} | Helped |
| Use LieNLSD instead of LieGAN for symmetry discovery | Symmetry discovery · Wave equation | Grassmann distance: 2.36 \pm 0.15 -> (1.40 \pm 0.01) \times 10^{-2} | Helped |
| Use LieNLSD instead of LieGAN for symmetry discovery | Symmetry discovery · Schrödinger equation | Grassmann distance: 2.22 \pm 0.05 -> (8.62 \pm 1.31) \times 10^{-2} | Helped |
| Add LieNLSD-identified LPSDA to FNO | FNO · Burgers' equation | long rollout test NMSE: (2.33 \pm 1.07) \times 10^{-4} -> (1.80 \pm 0.56) \times 10^{-4} | Helped |
| Add LieNLSD-identified LPSDA to FNO | FNO · Heat equation | long rollout test NMSE: (1.07 \pm 0.10) \times 10^{-1} -> (5.99 \pm 0.04) \times 10^{-2} | Helped |
| Add LieNLSD-identified LPSDA to FNO | FNO · KdV equation | long rollout test NMSE: (1.74 \pm 0.10) \times 10^{-1} -> (1.42 \pm 0.16) \times 10^{-1} | Helped |
| Use LieNLSD-based rather than prior-method-based LPSDA | FNO · Burgers' equation | long rollout test NMSE: (1.93 \pm 0.75) \times 10^{-4} -> (1.80 \pm 0.56) \times 10^{-4} | Helped |
| Use LieNLSD-based rather than prior-method-based LPSDA | FNO · Heat equation | long rollout test NMSE: (7.33 \pm 1.07) \times 10^{-2} -> (5.99 \pm 0.04) \times 10^{-2} | Helped |
| Use LieNLSD-based rather than prior-method-based LPSDA | FNO · KdV equation | long rollout test NMSE: (1.51 \pm 0.01) \times 10^{-1} -> (1.42 \pm 0.16) \times 10^{-1} | Helped |
| Use LieNLSD-based rather than ground-truth-based LPSDA | FNO · Heat equation | long rollout test NMSE: (6.01 \pm 0.20) \times 10^{-2} -> (5.99 \pm 0.04) \times 10^{-2} | No clear effect |
Limitations
- Function library must be specified manually, limiting the search space to chosen terms (up to second order in experiments).
- Symmetry discovery is restricted to Lie point symmetries acting pointwise; not symmetries over whole discretized fields.
- LPSDA application limited to one-dimensional cases due to its technical constraints.
Experimental setup
Datasets: top quark tagging Burgers' equation wave equation Schrödinger equation heat equation KdV equation
Benchmarks: top quark tagging Burgers' equation wave equation Schrödinger equation heat equation KdV equation
Metrics: Grassmann distance parameter count long rollout NMSE
Baselines: LieGAN LieGG L-conv LaLiGAN LPSDA with symmetries from the MLP-based nonlinear discovery method
Builds on: LieGAN LaLiGAN LieGG LADMAP Lie point symmetry data augmentation
Closest papers by meaning
- Learning Infinitesimal Generators of Continuous Symmetries from Data (NeurIPS 2024 · similarity 0.89)
- Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups (ICLR 2025 · similarity 0.88)
- Symmetry-Informed Governing Equation Discovery (NeurIPS 2024 · similarity 0.88)
- Discovering Symmetries of ODEs by Symbolic Regression (AAAI 2025 · similarity 0.87)
- Symmetry Discovery Beyond Affine Transformations (NeurIPS 2024 · similarity 0.86)
- AtlasD: Automatic Local Symmetry Discovery (ICML 2025 · similarity 0.86)
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