cs.LGMar 24, 2026

Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation

Authors: Xinxin Li, Xingyu Cui, Jin Qi, Juan Zhang, Da Li, Junping Yin

Organizations: School of Mathematical Sciences, East China Normal University, Shang Hai, 200241, China · Institute of Applied Physics and Computational Mathematics, 100094, Beijing, China · Shanghai Zhangjiang Institute of Mathematics, 201203, Shanghai, China · Institute of Artificial Intelligence, Beihang University, 100191, Beijing, China · Academy for Advanced Interdisciplinary Studies, Northeast Normal University, 130024, Changchun, Jilin, China

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Abstract

Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the instability of numerical differentiation under sparse and noisy data, and (ii) the restricted flexibility of a pre-defined candidate library. We propose Weak-PDE-Net, an end-to-end differentiable framework that can robustly identify open-form PDEs. Weak-PDE-Net consists of two interconnected modules: a forward response learner and a weak-form PDE generator. The learner embeds learnable Gaussian kernels within a lightweight MLP, serving as a surrogate model that adaptively captures system dynamics from sparse observations. Meanwhile, the generator integrates a symbolic network with an integral module to construct weak-form PDEs, avoiding explicit numerical differentiation and improving robustness to noise. To relax the constraints of the pre-defined library, we leverage Differentiable Neural Architecture Search strategy during training to explore the functional space, which enables the efficient discovery of open-form PDEs. The capability of Weak-PDE-Net in multivariable systems discovery is further enhanced by incorporating Galilean Invariance constraints and symmetry equivariance hypotheses to ensure physical consistency. Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.

Figures & tables

Figure 1: The overall architecture of Weak-PDE-Net.
Figure 2: Overview of the training process, exemplified by the Burgers equation. The training progress through two parallel tracks. The PDE Learning Process focuses on learning the PDE solution, consistently reconstructing the full grid data. Concurrently, the PDE Discovery Process evolves from symbolic network architecture adaptation, to connectivity pruning for structural sparsity, and finally to precise coefficient refinement. These two tracks are jointly optimized via a shared loss.
Figure 3: The forward propagation of the symbolic network during the searching phase. The left panel illustrates the overall forward propagation framework; the right panel provides the specific details of the functional operation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%2.5%1.000.05890.05572.97×10−42.97\times 10^{-4}∂tu=−0.4722∂xu2+0.1059∂xxu\partial_{t}u=-0.4722\partial_{x}u^{2}+0.1059\partial_{xx}u
0%5.0%1.000.01840.01821.54×10−41.54\times 10^{-4}∂tu=−0.4908∂xu2+0.0992∂xxu\partial_{t}u=-0.4908\partial_{x}u^{2}+0.0992\partial_{xx}u
0%10%1.000.00960.00957.14×10−57.14\times 10^{-5}∂tu=−0.5048∂xu2+0.1007∂xxu\partial_{t}u=-0.5048\partial_{x}u^{2}+0.1007\partial_{xx}u
0%25%1.000.02130.00475.07×10−55.07\times 10^{-5}∂tu=−0.5011∂xu2+0.1021∂xxu\partial_{t}u=-0.5011\partial_{x}u^{2}+0.1021\partial_{xx}u
0%50%1.000.00120.00084.84×10−54.84\times 10^{-5}∂tu=−0.4996∂xu2+0.1001∂xxu\partial_{t}u=-0.4996\partial_{x}u^{2}+0.1001\partial_{xx}u
Table 1: Performance evaluation on Burgers equation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%2.5%1.000.07190.02591.38×10−41.38\times 10^{-4}∂tu=−3.0394∂xu2−0.9281∂xxxu\partial_{t}u=-3.0394\partial_{x}u^{2}-0.9281\partial_{xxx}u
0%5.0%1.000.03730.01391.48×10−41.48\times 10^{-4}∂tu=−2.9766∂xu2−1.0373∂xxxu\partial_{t}u=-2.9766\partial_{x}u^{2}-1.0373\partial_{xxx}u
0%10%1.000.02540.01204.12×10−54.12\times 10^{-5}∂tu=−2.9718∂xu2−1.0254∂xxxu\partial_{t}u=-2.9718\partial_{x}u^{2}-1.0254\partial_{xxx}u
0%25%1.000.03760.01196.49×10−56.49\times 10^{-5}∂tu=−3.0013∂xu2−1.0376∂xxxu\partial_{t}u=-3.0013\partial_{x}u^{2}-1.0376\partial_{xxx}u
0%50%1.000.02740.01031.47×10−51.47\times 10^{-5}∂tu=−3.0174∂xu2−1.0274∂xxxu\partial_{t}u=-3.0174\partial_{x}u^{2}-1.0274\partial_{xxx}u
Table 2: Performance evaluation on the KdV equation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%2.5%1.000.01730.01494.47×10−34.47\times 10^{-3}∂tu=−0.4916∂x(u)2−1.0115∂xxu−1.0173∂xxxxu\partial_{t}u=-0.4916\partial_{x}(u)^{2}-1.0115\partial_{xx}u-1.0173\partial_{xxxx}u
0%5.0%1.000.00760.00452.13×10−32.13\times 10^{-3}∂tu=−0.4962∂x(u2)−0.9946∂xxu−0.9989∂xxxxu\partial_{t}u=-0.4962\partial_{x}(u^{2})-0.9946\partial_{xx}u-0.9989\partial_{xxxx}u
0%10%1.000.01200.00422.77×10−32.77\times 10^{-3}∂tu=−0.4940∂x(u2)−0.9998∂xxu−1.0012∂xxxxu\partial_{t}u=-0.4940\partial_{x}(u^{2})-0.9998\partial_{xx}u-1.0012\partial_{xxxx}u
0%25%1.000.00550.00211.66×10−31.66\times 10^{-3}∂tu=−0.5027∂x(u2)−0.9985∂xxu−0.9995∂xxxxu\partial_{t}u=-0.5027\partial_{x}(u^{2})-0.9985\partial_{xx}u-0.9995\partial_{xxxx}u
0%50%1.000.00320.00291.01×10−31.01\times 10^{-3}∂tu=−0.4998∂x(u2)−1.0032∂xxu−1.0030∂xxxxu\partial_{t}u=-0.4998\partial_{x}(u^{2})-1.0032\partial_{xx}u-1.0030\partial_{xxxx}u
Table 3: Performance evaluation on the KS equation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%2.5%1.000.21490.15239.25×10−49.25\times 10^{-4}∂tu=0.9298u3−0.7851u+0.8643∂xxu\partial_{t}u=0.9298u^{3}-0.7851u+0.8643\partial_{xx}u
0%5.0%1.000.15200.11891.93×10−51.93\times 10^{-5}∂tu=0.9200u3−0.8480u+0.8865∂xxu\partial_{t}u=0.9200u^{3}-0.8480u+0.8865\partial_{xx}u
0%10%1.000.11390.09682.37×10−52.37\times 10^{-5}∂tu=0.9314u3−0.8861u+0.8977∂xxu\partial_{t}u=0.9314u^{3}-0.8861u+0.8977\partial_{xx}u
0%25%1.000.12240.09556.06×10−66.06\times 10^{-6}∂tu=0.9317u3−0.8776u+0.9123∂xxu\partial_{t}u=0.9317u^{3}-0.8776u+0.9123\partial_{xx}u
0%50%1.000.11210.09261.00×10−51.00\times 10^{-5}∂tu=0.9297u3−0.8879u+0.9094∂xxu\partial_{t}u=0.9297u^{3}-0.8879u+0.9094\partial_{xx}u
Table 4: Performance evaluation on the CI equation. The symbol −−-- indicates that forward learning is not required under full sampling conditions; therefore, the reconstruction MSE is not applicable.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%10%1.001.00 1.001.000.01240.0124 0.01010.01010.00960.0096 0.00730.00731.64×10−41.64\times 10^{-4} 1.57×10−41.57\times 10^{-4}{∂tu=0.4955∂xxv+0.9876u2v+1.0056v3∂tv=−0.4993∂xxu−0.9899uv2−1.0041u3\begin{cases}\partial_{t}u=0.4955\partial_{xx}v+0.9876u^{2}v+1.0056v^{3}\\ \partial_{t}v=-0.4993\partial_{xx}u-0.9899uv^{2}-1.0041u^{3}\end{cases}
0%25%1.001.00 1.001.000.00490.0049 0.00340.00340.00340.0034 0.00250.00253.17×10−53.17\times 10^{-5} 2.69×10−52.69\times 10^{-5}{∂tu=0.4990∂xxv+0.9951u2v+1.0008v3∂tv=−0.4991∂xxu−1.0034uv2−0.9987u3\begin{cases}\partial_{t}u=0.4990\partial_{xx}v+0.9951u^{2}v+1.0008v^{3}\\ \partial_{t}v=-0.4991\partial_{xx}u-1.0034uv^{2}-0.9987u^{3}\end{cases}
0%37.5%1.001.00 1.001.000.00430.0043 0.00220.00220.00310.0031 0.00150.00151.21×10−51.21\times 10^{-5} 1.06×10−51.06\times 10^{-5}{∂tu=0.4988∂xxv+1.0043u2v+0.9986v3∂tv=−0.5001∂xxu−1.0022uv2−0.9995u3\begin{cases}\partial_{t}u=0.4988\partial_{xx}v+1.0043u^{2}v+0.9986v^{3}\\ \partial_{t}v=-0.5001\partial_{xx}u-1.0022uv^{2}-0.9995u^{3}\end{cases}
0%50%1.001.00 1.001.000.00410.0041 0.00240.00240.00210.0021 0.00170.00171.56×10−51.56\times 10^{-5} 1.12×10−51.12\times 10^{-5}{∂tu=0.4980∂xxv+0.9977u2v+0.9998v3∂tv=−0.4997∂xxu−0.9976uv2−1.0006u3\begin{cases}\partial_{t}u=0.4980\partial_{xx}v+0.9977u^{2}v+0.9998v^{3}\\ \partial_{t}v=-0.4997\partial_{xx}u-0.9976uv^{2}-1.0006u^{3}\end{cases}
Varying Noise Level ( r=25%r=25\% )
Table 5: Performance evaluation on the NLS equation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%1.0%1.000.01810.01357.12×10−37.12\times 10^{-3}∂ttu=0.9943∂xxu+0.9819∂yyu\partial_{tt}u=0.9943\partial_{xx}u+0.9819\partial_{yy}u
0%2.5%1.000.01580.01338.35×10−48.35\times 10^{-4}∂ttu=0.9842∂xxu+0.9899∂yyu\partial_{tt}u=0.9842\partial_{xx}u+0.9899\partial_{yy}u
0%5.0%1.000.00800.00569.19×10−49.19\times 10^{-4}∂ttu=0.9999∂xxu+0.9920∂yyu\partial_{tt}u=0.9999\partial_{xx}u+0.9920\partial_{yy}u
0%10%1.000.00410.00347.16×10−47.16\times 10^{-4}∂ttu=1.0024∂xxu+1.0041∂yyu\partial_{tt}u=1.0024\partial_{xx}u+1.0041\partial_{yy}u
0%25%1.000.00520.00507.48×10−47.48\times 10^{-4}∂ttu=1.0052∂xxu+1.0048∂yyu\partial_{tt}u=1.0052\partial_{xx}u+1.0048\partial_{yy}u
Table 6: Performance evaluation on the 2D Wave equation.
σNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Varying Sample Ratio
0%2.5%1.000.30040.18001.02×10−21.02\times 10^{-2}∂ttu=0.6996∂xxu+0.9995∂yyu−1.0839sin⁡(u)\partial_{tt}u=0.6996\partial_{xx}u+0.9995\partial_{yy}u-1.0839\sin(u)
0%5.0%1.000.26080.18049.68×10−39.68\times 10^{-3}∂ttu=0.8280∂xxu+0.9965∂yyu−0.7392sin⁡(u)\partial_{tt}u=0.8280\partial_{xx}u+0.9965\partial_{yy}u-0.7392\sin(u)
0%10%1.000.13410.10523.08×10−33.08\times 10^{-3}∂ttu=0.8775∂xxu+0.9855∂yyu−1.1341sin⁡(u)\partial_{tt}u=0.8775\partial_{xx}u+0.9855\partial_{yy}u-1.1341\sin(u)
0%25%1.000.17720.10263.21×10−33.21\times 10^{-3}∂ttu=1.0049∂xxu+0.9875∂yyu−0.8228sin⁡(u)\partial_{tt}u=1.0049\partial_{xx}u+0.9875\partial_{yy}u-0.8228\sin(u)
0%50%1.000.16890.09763.95×10−33.95\times 10^{-3}∂ttu=1.0042∂xxu+1.0030∂yyu−1.1689sin⁡(u)\partial_{tt}u=1.0042\partial_{xx}u+1.0030\partial_{yy}u-1.1689\sin(u)
Table 7: Performance evaluation on the SG equation.
ReReσNR\sigma_{\text{NR}}rrTPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
100Varying Sample Ratio ( σNR=0%\sigma_{\text{NR}}=0\% )
0%5%1.000.03740.01411.35×10−31.35\times 10^{-3}∂tω=−0.9824∂x(uω)−0.9904∂y(vω)+0.0097∂xxω+0.0104∂yyω\partial_{t}\omega=-0.9824\partial_{x}(u\omega)-0.9904\partial_{y}(v\omega)+0.0097\partial_{xx}\omega+0.0104\partial_{yy}\omega
0%10%1.000.01700.00521.23×10−31.23\times 10^{-3}∂tω=−0.9927∂x(uω)−1.0004∂y(vω)+0.0099∂xxω+0.0102∂yyω\partial_{t}\omega=-0.9927\partial_{x}(u\omega)-1.0004\partial_{y}(v\omega)+0.0099\partial_{xx}\omega+0.0102\partial_{yy}\omega
0%100%1.000.00510.0043−−--∂tω=−1.0010∂x(uω)−0.9949∂y(vω)+0.0100∂xxω+0.0100∂yyω\partial_{t}\omega=-1.0010\partial_{x}(u\omega)-0.9949\partial_{y}(v\omega)+0.0100\partial_{xx}\omega+0.0100\partial_{yy}\omega
Varying Noise Level ( r=10%r=10\% )
10%10%1.000.09720.02049.51×10−49.51\times 10^{-4}∂tω=−0.9792∂x(uω)−0.9800∂y(vω)+0.0110∂xxω+0.0098∂yyω\partial_{t}\omega=-0.9792\partial_{x}(u\omega)-0.9800\partial_{y}(v\omega)+0.0110\partial_{xx}\omega+0.0098\partial_{yy}\omega
Table 8: Performance evaluation on the incompressible NS equation at Re=100Re=100 . The symbol −−-- indicates that forward learning is not required under full sampling conditions; therefore, the reconstruction MSE is not applicable.
Figure 4: The results of the ablation studies. (a) A radar chart illustrating the True Positive Rate TPR . (b) and (c) Line plots showcasing the parameter errors E∞E_{\infty} and E2E_{2} , respectively. (d) A bar chart representing the reconstructed error L(U^,U)\mathcal{L}(\hat{U},U) .
NoiseBurgersKdVKS
E∞E_{\infty}E2E_{2}E∞E_{\infty}E2E_{2}E∞E_{\infty}E2E_{2}
WPLOursWPLOursWPLOursWPLOursWPLOursWPLOurs
25%0.01300.00980.01300.00970.00700.00400.00360.00360.01000.00660.00940.0049
50%0.08080.05930.08010.01170.02620.01360.02590.00620.03830.04870.04000.0342
75%0.15300.17460.09820.03430.05200.03930.04940.03520.11530.08950.13000.0331
100%0.34000.19310.13340.05520.18380.07320.17520.0719–0.1340–0.0973
Table 9: Comparative evaluation against Weak-PDE-LEARN (WPL). The symbol – for E∞E_{\infty} and E2E_{2} indicates that the equation was not correctly identified ( TPR<1\text{TPR}<1 ).
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter NameSymbolValue
Test Function TypeΦ\PhiPoly-bump (KdV) / Exp-bump (Others)
Support Radius Range(ηmin,ηmax)(\eta_{\text{min}},\eta_{\text{max}})(0.2,0.4)(0.2,0.4)
Number of Gaussian KernelsNN10001000
Gaussian Feature Dimensiond′d^{\prime}44
Lightweight MLP DepthLL22
MLP Activation Functionσ\sigmasin⁡\sin (NS) / tanh⁡\tanh (Others)
Appendix
Table 10: Hyperparameter settings used in the experiments.
Table 11: Reconstruction performance of the learned response function UU for 1D equations.
Table 12: Reconstruction performance of the learned response function UU for 2D equations.
Table 13: Reconstruction performance of the learned response function UU for 2D NS equation.
EquationSettingsTPR{\mathrm{TPR}}E∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Burgersr=10%r=10\% , FEϕ\mathrm{FE}_{\phi} +NAS1.000.00960.00957.14×10−57.14\times 10^{-5}∂tu=−0.5048∂xu2+0.1007∂xxu\partial_{t}u=-0.5048\partial_{x}u^{2}+0.1007\partial_{xx}u
r=10%r=10\% , without FEϕ\mathrm{FE}_{\phi}0.001.00003.19282.30×10−42.30\times 10^{-4}∂tu=−1.5006∂xu4+0.3724∂xxu4\partial_{t}u=-1.5006\partial_{x}u^{4}+0.3724\partial_{xx}u^{4}
r=10%r=10\% , without NAS0.172.19148.97318.63×10−58.63\times 10^{-5}∂tu=3.0014∂xu4−3.2493∂xu3+0.5957∂xu2+⋯\partial_{t}u=3.0014\partial_{x}u^{4}-3.2493\partial_{x}u^{3}+0.5957\partial_{x}u^{2}+\cdots
KdVr=25%r=25\% , FEϕ\mathrm{FE}_{\phi} +NAS1.000.03760.01196.49×10−56.49\times 10^{-5}∂tu=−3.0013∂xu2−1.0376∂xxxu\partial_{t}u=-3.0013\partial_{x}u^{2}-1.0376\partial_{xxx}u
r=25%r=25\% , without FEϕ\mathrm{FE}_{\phi}0.001.00001.13797.55×10−37.55\times 10^{-3}∂t=0.3366u2−0.4742∂xu+1.5952∂xxu2−0.2539∂xxu\partial_{t}=0.3366u^{2}-0.4742\partial_{x}u+1.5952\partial_{xx}u^{2}-0.2539\partial_{xx}u
r=25%r=25\% , without NAS0.330.73690.75982.90×10−32.90\times 10^{-3}∂t=−0.7894∂xu2−0.6544∂xxxu+⋯\partial_{t}=-0.7894\partial_{x}u^{2}-0.6544\partial_{xxx}u+\cdots
Appendix
Table 14: Detailed results of ablation studies.
σNR\sigma_{\text{NR}}NdataN_{\text{data}}TPRE∞{E_{\infty}}E2{E_{2}}L(U^,U){\mathcal{L}(\hat{U},U)}Discovered Equation
Burgers Equation
25%40001.000.00980.00972.67×10−42.67\times 10^{-4}∂tu=−0.4950∂xu2+0.0999∂xxu\partial_{t}u=-0.4950\partial_{x}u^{2}+0.0999\partial_{xx}u
50%40001.000.05930.01177.58×10−47.58\times 10^{-4}∂tu=−0.4992∂xu2+0.0941∂xxu\partial_{t}u=-0.4992\partial_{x}u^{2}+0.0941\partial_{xx}u
75%40001.000.17460.03432.35×10−32.35\times 10^{-3}∂tu=−0.5010∂xu2+0.1175∂xxu\partial_{t}u=-0.5010\partial_{x}u^{2}+0.1175\partial_{xx}u
100%40001.000.19310.05524.24×10−34.24\times 10^{-3}∂tu=−0.5011∂xu2+0.1021∂xxu\partial_{t}u=-0.5011\partial_{x}u^{2}+0.1021\partial_{xx}u
KdV Equation
Appendix
Table 15: Detailed results of our method in comparison experiments.

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