cs.SCSep 27, 2026

LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery

Authors: Xinxin Li, Jianming Ma, Xingyu Cui, Da Li, Juan Zhang, Junping Yin

Organizations: East China Normal University · Shanghai Jiao Tong University · Institute of Applied Physics and Computational Mathematics · Northeast Normal University · Beihang University

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Abstract

Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.

Figures & tables

Figure 1: The overall pipeline of LieDiscover. On the left, symmetry discovery is formulated as an MDP, using solid arrows for information flow and dashed arrows for training network. On the right, the extracted symmetry generators are applied to downstream PDE tasks, such as data augmentation and equation discovery.
MethodTopBurg.HeatDampedForced
LieGAN(2.51±0.41)×10−1(2.51\pm 0.41)\times 10^{-1}1.58±0.051.58\pm 0.052.59±0.042.59\pm 0.042.20±0.012.20\pm 0.011.83±0.091.83\pm 0.09
LieNLSD(1.24±0.17)×10−1(1.24\pm 0.17)\times 10^{-1}(2.29±0.56)×10−2(2.29\pm 0.56)\times 10^{-2}(8.90±2.94)×10−4(8.90\pm 2.94)\times 10^{-4}1.57±0.001.57\pm 0.00(1.12±0.75)×10−2(1.12\pm 0.75)\times 10^{-2}
LieDiscover(8.52±0.39)×10−2\mathbf{(8.52\pm 0.39)\times 10^{-2}}(1.76±0.07)×10−2\mathbf{(1.76\pm 0.07)\times 10^{-2}}(5.04±2.72)×10−4\mathbf{(5.04\pm 2.72)\times 10^{-4}}(1.94±0.81)×10−2\mathbf{(1.94\pm 0.81)\times 10^{-2}}(9.67±0.52)×10−3\mathbf{(9.67\pm 0.52)\times 10^{-3}}
Table 1: Grassmann distance between the discovered Lie algebra subspace and the ground-truth subspace. Results are reported as mean ±\pm standard deviation over three random seeds. The best result in each column is highlighted in bold.
DatasetHeatDampedForcedWave
Generatorsv1=(2t+x2)∂u,v2=2t∂t+x∂x,v3=2t∂x−xu∂u,v4=x∂u, v5=u∂u,v6=∂u, v7=∂x,v8=∂t,v9=(6tx+x3)∂u.\begin{aligned} \mathbf{v}_{1}&=(2t+x^{2})\partial_{u},\\ \mathbf{v}_{2}&=2t\partial_{t}+x\partial_{x},\\ \mathbf{v}_{3}&=2t\partial_{x}-xu\partial_{u},\\ \mathbf{v}_{4}&=x\partial_{u},\ \mathbf{v}_{5}=u\partial_{u},\\ \mathbf{v}_{6}&=\partial_{u},\ \mathbf{v}_{7}=\partial_{x},\\ \mathbf{v}_{8}&=\partial_{t},\\ \mathbf{v_{9}}&=(6tx+x^{3})\partial_{u}.\end{aligned}v1=∂t,v2=∂x,v3=e−t∂x −e−t∂u.\begin{aligned} \mathbf{v}_{1}&=\partial_{t},\\ \mathbf{v}_{2}&=\partial_{x},\\ \mathbf{v}_{3}&=e^{-t}\partial_{x}\\ &\ -e^{-t}\partial_{u}.\end{aligned}v1=∂t, v2=∂u,v3=∂x+sin⁡(x)∂u,v4=u∂u+cos⁡(x)∂u,v5=(t−x)∂u,v6=sin⁡(t−x)∂u,v7=cos⁡(t−x)∂u,v8=(t−x)∂t,v9=sin⁡(t−x)∂t,v10=cos⁡(t−x)∂t.\begin{aligned} \mathbf{v}_{1}&=\partial_{t},\ \mathbf{v}_{2}=\partial_{u},\\ \mathbf{v}_{3}&=\partial_{x}+\sin(x)\partial_{u},\\ \mathbf{v}_{4}&=u\partial_{u}+\cos(x)\partial_{u},\\ \mathbf{v}_{5}&=(t-x)\partial_{u},\\ \mathbf{v}_{6}&=\sin(t-x)\partial_{u},\\ \mathbf{v}_{7}&=\cos(t-x)\partial_{u},\\ \mathbf{v}_{8}&=(t-x)\partial_{t},\\ \mathbf{v}_{9}&=\sin(t-x)\partial_{t},\\ \mathbf{v}_{10}&=\cos(t-x)\partial_{t}.\end{aligned}v1=2tx∂t+(t2+x2−y2)∂x+2xy∂y−xu∂u,v2=2ty∂t+2xy∂x+(t2−x2+y2)∂y−yu∂u,v3=x∂t+t∂x, v4=y∂t+t∂y,v5=t∂t+x∂x+y∂y, v6=−y∂x+x∂y,v7=u∂u, v8=∂t, v9=∂x, v10=∂y,v11=∂u, v12=t∂u, v13=x∂u, v14=y∂u,v15=tx∂u, v16=ty∂u, v17=xy∂u, v18=txy∂u,v19=(t2+y2)∂u, v20=(t2+2x2−y2)∂u,v21=(t3+3tx2)∂u, v22=(t3+3ty2)∂u.\begin{aligned} \mathbf{v}_{1}&=2tx\partial_{t}+(t^{2}+x^{2}-y^{2})\partial_{x}+2xy\partial_{y}-xu\partial_{u},\\ \mathbf{v}_{2}&=2ty\partial_{t}+2xy\partial_{x}+(t^{2}-x^{2}+y^{2})\partial_{y}-yu\partial_{u},\\ \mathbf{v}_{3}&=x\partial_{t}+t\partial_{x},\ \mathbf{v}_{4}=y\partial_{t}+t\partial_{y},\\ \mathbf{v}_{5}&=t\partial_{t}+x\partial_{x}+y\partial_{y},\ \mathbf{v}_{6}=-y\partial_{x}+x\partial_{y},\\ \mathbf{v}_{7}&=u\partial_{u},\ \mathbf{v}_{8}=\partial_{t},\ \mathbf{v}_{9}=\partial_{x},\ \mathbf{v}_{10}=\partial_{y},\\ \mathbf{v}_{11}&=\partial_{u},\ \mathbf{v}_{12}=t\partial_{u},\ \mathbf{v}_{13}=x\partial_{u},\ \mathbf{v}_{14}=y\partial_{u},\\ \mathbf{v}_{15}&=tx\partial_{u},\ \mathbf{v}_{16}=ty\partial_{u},\ \mathbf{v}_{17}=xy\partial_{u},\ \mathbf{v}_{18}=txy\partial_{u},\\ \mathbf{v}_{19}&=(t^{2}+y^{2})\partial_{u},\ \mathbf{v}_{20}=(t^{2}+2x^{2}-y^{2})\partial_{u},\\ \mathbf{v}_{21}&=(t^{3}+3tx^{2})\partial_{u},\ \mathbf{v}_{22}=(t^{3}+3ty^{2})\partial_{u}.\end{aligned}
Table 2: Infinitesimal generators discovered by LieDiscover on selected PDEs.
DatasetFNO + ∅\emptysetFNO + Ko et al.FNO + LieNLSDFNO + LieDiscoverFNO + GT
Burg.(2.33±1.07)×10−4(2.33\pm 1.07)\times 10^{-4}(1.93±0.75)×10−4(1.93\pm 0.75)\times 10^{-4}(1.31±0.12)×10−4(1.31\pm 0.12)\times 10^{-4}(1.24±0.33)×10−4\mathbf{(1.24\pm 0.33)\times 10^{-4}}(1.75±0.37)×10−4(1.75\pm 0.37)\times 10^{-4}
Heat(1.07±0.10)×10−1(1.07\pm 0.10)\times 10^{-1}(7.33±1.07)×10−2(7.33\pm 1.07)\times 10^{-2}(5.25±0.31)×10−2(5.25\pm 0.31)\times 10^{-2}(5.02±0.10)×10−2\mathbf{(5.02\pm 0.10)\times 10^{-2}}(6.01±0.20)×10−2(6.01\pm 0.20)\times 10^{-2}
KdV(1.74±0.10)×10−1(1.74\pm 0.10)\times 10^{-1}(1.51±0.01)×10−1(1.51\pm 0.01)\times 10^{-1}(1.27±0.12)×10−1(1.27\pm 0.12)\times 10^{-1}(1.26±0.14)×10−1\mathbf{(1.26\pm 0.14)\times 10^{-1}}(1.47±0.02)×10−1(1.47\pm 0.02)\times 10^{-1}
Damped(2.97±0.52)×10−3(2.97\pm 0.52)\times 10^{-3}(2.91±0.49)×10−3(2.91\pm 0.49)\times 10^{-3}(2.51±0.00)×10−3(2.51\pm 0.00)\times 10^{-3}(1.78±0.06)×10−3\mathbf{(1.78\pm 0.06)\times 10^{-3}}(1.90±0.08)×10−3(1.90\pm 0.08)\times 10^{-3}
Forced(1.65±0.07)×10−1(1.65\pm 0.07)\times 10^{-1}(1.68±0.07)×10−1(1.68\pm 0.07)\times 10^{-1}(1.61±0.01)×10−1(1.61\pm 0.01)\times 10^{-1}(1.59±0.02)×10−1(1.59\pm 0.02)\times 10^{-1}(1.24±0.02)×10−1\mathbf{(1.24\pm 0.02)\times 10^{-1}}
Table 3: Comparison of long-rollout test NMSE for FNO under under different data augmentations. The best result is highlighted.
Figure 2: Performance of different PDE discovery methods. The three parts show TPR ( ↑\uparrow ), E∞E_{\infty} ( ↓\downarrow ), and E2E_{2} ( ↓\downarrow ), respectively.
Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

 ({Wxi},Wuα)←\textscRecursiveCoeff(uα,J)(\{W_{x_{i}}\},W_{u^{\alpha}})\leftarrow\textsc{RecursiveCoeff}(u^{\alpha},J) 
Appendix
Algorithm 1 Recursive Construction of Θn\Theta_{n} from Θ\Theta
 G=C⊤C=∑i=1MCi⊤Ci.G=C^{\top}C=\sum_{i=1}^{M}C_{i}^{\top}C_{i}. 
Appendix
Algorithm 3 Adaptive SVD
 Pk−Q⊤(Λk+βk(QPk−Zk))βkηP.P_{k}-\frac{Q^{\top}(\Lambda_{k}+\beta_{k}(QP_{k}-Z_{k}))}{\beta_{k}\eta_{P}}. 
Appendix
Algorithm 4 Basis Sparsification and Pruning
Figure 3: Detailed Jacobian computation steps.
MethodKdVWaveReaction-diffusionSchrödinger
LieGAN1.56±0.00031.56\pm 0.00032.36±0.152.36\pm 0.151.11±0.111.11\pm 0.112.22±0.052.22\pm 0.05
LieNLSD(4.86±1.68)×10−3(4.86\pm 1.68)\times 10^{-3}(1.43±0.03)×10−2{(1.43\pm 0.03)\times 10^{-2}}(1.69±0.19)×10−1(1.69\pm 0.19)\times 10^{-1}(1.14±0.28)×10−1(1.14\pm 0.28)\times 10^{-1}
LieDiscover(4.09±2.63)×10−3\mathbf{(4.09\pm 2.63)\times 10^{-3}}(1.43±0.02)×10−2\mathbf{(1.43\pm 0.02)\times 10^{-2}}(1.65±0.29)×10−1\mathbf{(1.65\pm 0.29)\times 10^{-1}}(9.76±1.20)×10−2\mathbf{(9.76\pm 1.20)\times 10^{-2}}
Appendix
Table 4: Grassmann distance between the discovered Lie algebra subspace and the ground-truth subspace. The results are reported as mean ±\pm standard deviation over three random seeds. Lower values are better.
Figure 4: Visualization result of symmetry discovery on top quark tagging by LieDiscover. The first subplot shows the last 8 singular values. The other seven subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 5: Visualization result of symmetry discovery on Burgers’ Equation by LieDiscover. The first subplot shows the last 7 singular values. The other six subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 6: Visualization result of symmetry discovery on Heat Equation by LieDiscover. The first subplot shows the last 10 singular values. The other nine subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 7: Visualization result of symmetry discovery on KdV Equation by LieDiscover. The first subplot shows the last 5 singular values. The other four subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 8: Visualization result of symmetry discovery on Damped Burgers’ Equation by LieDiscover. The first subplot shows the last 4 singular values. The other three subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 9: Visualization result of symmetry discovery on Forced Transport Equation by LieDiscover. The first subplot shows the last 11 singular values. The other ten subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 10: Visualization result of symmetry discovery on Wave Equation by LieDiscover. The first subplot shows the last 23 singular values. The other 22 subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 11: Visualization result of symmetry discovery on Schrödinger Equation by LieDiscover. The first subplot shows the last 7 singular values. The other six subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
Figure 12: Visualization result of symmetry discovery on Reaction-diffusion Equation by LieDiscover. The first subplot shows the last 6 singular values. The other five subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
DataMethodTPR ↑\uparrowE∞↓E_{\infty}\downarrowE2↓E_{2}\downarrow
Burgers’EQL(3.17±0.22)×10−1(3.17\pm 0.22)\times 10^{-1}(1.64±0.05)×10−2\mathbf{(1.64\pm 0.05)\times 10^{-2}}(6.43±1.35)×10−1(6.43\pm 1.35)\times 10^{-1}
 LieEQL1.00±0.00\mathbf{1.00\pm 0.00}(1.97±0.00)×10−2(1.97\pm 0.00)\times 10^{-2}(1.91±0.00)×10−2(1.91\pm 0.00)\times 10^{-2}
 SINDy1.00±0.001.00\pm 0.00(1.99±0.12)×10−2(1.99\pm 0.12)\times 10^{-2}(1.94±0.11)×10−2(1.94\pm 0.11)\times 10^{-2}
 LieSINDy1.00±0.001.00\pm 0.00(1.90±0.11)×10−2(1.90\pm 0.11)\times 10^{-2}(1.80±0.11)×10−2\mathbf{(1.80\pm 0.11)\times 10^{-2}}
HeatEQL1.00±0.00\mathbf{1.00\pm 0.00}(2.01±0.00)×10−2(2.01\pm 0.00)\times 10^{-2}(2.01±0.00)×10−2(2.01\pm 0.00)\times 10^{-2}
 LieEQL1.00±0.001.00\pm 0.00(2.02±0.02)×10−2(2.02\pm 0.02)\times 10^{-2}(2.02±0.02)×10−2(2.02\pm 0.02)\times 10^{-2}
Appendix
Table 5: Quantitative comparison of PDE discovery methods on all datasets.

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