Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

Xingyu Chen; Junxiu An; Jun Guo; Yuqian Zhou

cs.LGMar 23, 2026

Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

Authors: Xingyu Chen, Junxiu An, Jun Guo, Yuqian Zhou

Organizations: School of Software Engineering, Chengdu University of Information Technology, Chengdu, 610225, China · School of Statistics, Chengdu University of Information Technology, Chengdu, 610103, China · College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, China · Key Laboratory of Mathematical Meteorology Chengdu University of Information Technology Chengdu, 610225, China

Abstract

Data-driven discovery of partial differential equations (PDEs) offers a promising paradigm for uncovering governing physical laws from observational data. However, in practical scenarios, measurements are often contaminated by noise and limited by sparse sampling, which poses significant challenges to existing approaches based on numerical differentiation or integral formulations. In this work, we propose a Symbolic Graph Network (SGN) framework for PDE discovery under noisy and sparse conditions. Instead of relying on local differential approximations, SGN leverages graph message passing to model spatial interactions, providing a non-local representation that is less sensitive to high frequency noise. Based on this representation, the learned latent features are further processed by a symbolic regression module to extract interpretable mathematical expressions. We evaluate the proposed method on several benchmark systems, including the wave equation, convection-diffusion equation, and incompressible Navier-Stokes equations. Experimental results show that SGN can recover meaningful governing relations or solution forms under varying noise levels, and demonstrates improved robustness compared to baseline methods in sparse and noisy settings. These results suggest that combining graph-based representations with symbolic regression provides a viable direction for robust data-driven discovery of physical laws from imperfect observations. The code is available at https://github.com/CXY0112/SGN

Figures & tables

Figure 1: The overall framework of SGN. The architecture comprises two synergistic components: (1) the Graph Neural Simulator , which functions as a mesh-free discretization engine. It leverages graph message passing to extract robust non-local spatial operators from sparse and noisy observations, aided by auxiliary stabilization strategies; and (2) the Symbolic Inverse Analyzer , which distills these physics-aligned latent features into explicit, parsimonious governing equations.

Figure 2: Savitzky-Golay filtering. The dashed line exhibits the severe high-frequency jitter of raw noisy observations. In contrast, the solid curves demonstrate the smoothed physical trends recovered by increasing window sizes. This filtering effectively suppresses stochastic noise while preserving geometric structure, providing a stable, derivative-friendly initialization for the GNN simulator.

Noise Level	Method	MSE	Extracted Formula (Simplified)
0.0%	PDE-Net 2.0	$5.91\times 10^{-5}$	$\begin{aligned} u_{t}&=0.998v-0.099u\\ v_{t}&=\mathbf{-9.82u}-0.102v+1.17\times 10^{-5}(u_{xx}+u_{yy})\end{aligned}$
0.0%	SGN (Ours)	$2.54\times 10^{-4}$	$u\approx\sin(3.08x)\sin(3.10y)\cos(t/-0.317)$
0.1%	PDE-Net 2.0	$2.92\times 10^{-5}$	$\begin{aligned} u_{t}&=0.996v-0.092u\\ v_{t}&=\mathbf{-9.79u}-0.114v+7.32\times 10^{-6}(u_{xx}+u_{yy})\end{aligned}$
0.1%	SGN (Ours)	$5.41\times 10^{-4}$	$u\approx\sin(3.14x)\sin(y+\dots)\cos(-3.14t)/\sin(1.23)$
0.2%	PDE-Net 2.0	NaN	$\begin{aligned} u_{t}&=0.507v-0.011u_{y}+0.007u_{x}\\ v_{t}&=-9.10u+0.108-0.050(u_{xx}v)-0.018(u_{yy}u_{xx})\end{aligned}$
0.2%	SGN (Ours)	$1.38\times 10^{-10}$	$u=\mathbf{\sin(3.1416x)\sin(3.1416y)\cos(3.1416t)}$

Table 1: Comparison of discovered formulas for the Wave Equation across different noise levels. (Note: Constants in the extracted formulas are rounded to appropriate significant figures, and expressions are moderately simplified for readability. The complete, unedited formulas and detailed variable nomenclature are provided in the Appendix.)

Figure 3: Visual comparison of predicted wave fields ( $u(x,y,t)$ ) at time $t=1.0$ under different noise regimes. Top row: low noise (0.2%); Bottom row: severe noise (5.0%). Columns present Ground Truth, SGN (ours), and PDE-NET 2.0 (baseline). While all methods converge under low noise, SGN maintains structural stability and captures the physical wave front even at 5.0% noise, whereas PDE-NET 2.0 exhibits catastrophic numerical divergence (indicated by ‘Inf’ on a blank background).

Noise Level	Method	MSE (on True)	Extracted Evolution / Node Formula (Simplified)
0.0%	PDE-Net 2.0	$5.38\times 10^{-2}$	$u_{t}\approx\mathbf{0.96u}-0.71u_{x}-1.42u_{y}+0.016u_{xx}+0.024u_{yy}$
0.0%	SGN (Ours)	$5.56\times 10^{-3}$	$u\approx u_{\text{prev}}-0.004\big[u_{x}-u_{\text{prev}}+0.50u_{y}+\dots\big]$
0.1%	PDE-Net 2.0	$5.36\times 10^{-2}$	$u_{t}\approx\mathbf{1.35u}-0.71u_{x}-1.42u_{y}+0.013u_{xx}+0.020u_{yy}+0.010u_{xy}$
0.1%	SGN (Ours)	$1.68\times 10^{-3}$	$u\approx u_{\text{prev}}-0.003\big[u_{x}+0.61u_{y}-(u_{x}-u_{\text{prev}})^{2}+\dots\big]$
0.5%	PDE-Net 2.0	$4.03\times 10^{-2}$	$u_{t}\approx 0.83u-0.71u_{x}-1.42u_{y}\mathbf{+0.037(u_{x})^{2}}+0.014u_{xx}+0.032u_{yy}$
0.5%	SGN (Ours)	$1.43\times 10^{-3}$	$u\approx u_{\text{prev}}-0.004\big[u_{x}+0.51u_{y}-u_{\text{prev}}\big]$

Table 2: Comparison of discovered formulas for the Convection-Diffusion Equation across different noise levels. (Note: Constants in SGN’s extracted node formulas are moderately simplified for readability, with the global multiplier representing the learned time step

\Delta t

. The complete, unedited formulas and detailed variable nomenclature are provided in the Appendix.)

Figure 4: Visual comparison of predicted solutions for the Convection-Diffusion Equation with 0.5% noise. Spatial distributions are shown at two time steps: $t=0.2$ (top row) and $t=2.0$ (bottom row). Columns display Ground Truth, SGN, and PDE-NET 2.0. The figure illustrates the relative degradation of prediction coherent structures at longer simulation times for both learned models compared to Ground Truth.

Noise	MSE (on True)	Node Formula ( $u,v$ update rules)	Message Formula ( $e_{0},e_{1}$ spatial coupling)
0.0%	$9.69\times 10^{-4}$	$u\approx\mathbf{\cos(x)\sin(y)}\cdot\big[-0.61\sin(-0.13t)-0.98\big]$	[Decoupled: Messages ignored by node formula]
0.0%	$9.69\times 10^{-4}$	$v\approx\mathbf{-\cos(y)}\cdot(u_{x}+u_{y}-\sin(x))\cdot\big[0.94\cos(\dots)\big]$	[Decoupled: Messages ignored by node formula]
0.1%	$6.27\times 10^{-4}$	$u\approx u_{1}\cdot\big(\mathbf{0.9900558}-0.0016\cdot e_{0}\big)$	$e_{0}\approx 0.015(v_{1}-u_{2})$
0.1%	$6.27\times 10^{-4}$	$v\approx v_{1}\cdot\big(\mathbf{0.9902582}-0.0007\cdot e_{1}\big)$	$e_{1}\approx 0.159-0.026(t-u_{2})$
0.5%	$6.30\times 10^{-4}$	$u\approx\mathbf{0.9900131}\cdot u_{1}-0.0046\cdot e_{1}$	$e_{0}\approx 0.237-0.212\cos(u_{1}-v_{1}-u_{y1})$
0.5%	$6.30\times 10^{-4}$	$v\approx\mathbf{0.9900553}\cdot v_{1}+0.0030\cdot e_{1}$	$e_{1}\approx-0.005(u_{y1}+v_{1})$

Table 3: Discovery results for the Navier-Stokes (Taylor-Green Vortex) system across different noise levels. The intermediate graph message functions are represented by

e_{0}

and

e_{1}

. Constants are moderately simplified for readability. The complete, unedited formulas and detailed variable nomenclature are provided in the Appendix.)

Figure 5: Snapshots of the Navier-Stokes flow dynamics under a severe 10% noise level. The figure illustrates the Ground Truth alongside the SGN predictions. Even when the underlying data is heavily obscured by high-frequency artifacts, the proposed SGN architecture flawlessly maintains the structural stability of the fluid flow, achieving highly consistent physical predictions without relying on dense spatial sampling.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Variable	Physical and Architectural Description
t	The temporal evolution step $t$ of the dynamical system.
x , y	The spatial coordinates of the grid nodes.
u_prev	For scalar fields (Wave, Convection-Diffusion), the field value $u$ of the central node at the previous time step.
x_prev , y_prev	For vector fields (Navier-Stokes), the two state components of the central node at the previous time step, corresponding to horizontal ( $u$ ) and vertical ( $v$ ) velocities.
*_prev1	In message functions, the historical state variable of the neighbor node (source node, $j$ ).
*_prev2	In message functions, the historical state variable of the central node (target node, $i$ ).

Appendix

Table 1: Nomenclature and physical interpretations of the symbolic variables discovered by SGN.

Noise	MSE(Noise)	MSE(Original)	Type	Equation
0.0%	/	2.5431E-04	Msg	sin(sin(sin(sin(cos(((t - t) + -0.56922716) + sin(((-1.8829098 - (t + ux2)) * -0.119090885) - u_prev1))) + -1.2227476))) + -0.0041820854
0.0%	/	2.5431E-04	Node	(-0.0049376567 + (sin(y * 3.0965688) * sin(x * 3.084605))) * cos(t / -0.31685314)
0.1%	5.4111E-04	5.4102E-04	Msg	u_prev2(2.405846 - u_prev1)(u_prev1(-0.14979462) + ux2(-0.017480928) - (-1)0.02454977/(t - 10.5184039))
0.1%	5.4111E-04	5.4102E-04	Node	sin(sin(x3.1401417)cos(t(-3.1437826)))sin(y + y + y/0.8408427)/sin(1.2261999)
0.2%	4.7517E-07	1.3837E-10	Msg	((cos(cos(sin(u_prev1 + sin(u_prev1 + u_prev2)) + ((uy1 + (ux1 - t)) * 0.20625228))) * cos(t)) - (u_prev2 - -0.08223844)) * ((u_prev1 - 1.3154284) * -0.19282961)
0.2%	4.7517E-07	1.3837E-10	Node	sin(x3.1416032)sin(y3.1416051)cos(t*3.1416218)

Appendix

Table 2: Unedited symbolic expressions discovered for the Wave Equation across varying noise levels.

Noise	MSE(Noise)	MSE(Original)	Type	Equation
0.0%	/	5.5649E-03	Msg1	(0.26722786 - 0.37149414u_prev2)sin(u_prev2*(ux1 + 2.8515506))
			Msg2	ux2(sin(ux2)0.04038851 + 0.02046432)
			Node	u_prev + ((ux - ((u_prev + (((uy * -0.08273945) - 0.50470436) * uy)) - (sin(u_prev) * ((u_prev * t) + -1.7263777)))) * -0.0040644426)
0.1%	1.6789E-03	1.6792E-03	Msg1	(cos(uy2) - cos(uyy1(-0.0684862)))(cos(uyy2(-0.075827084)) + 0.92147017)(-0.10752371)
			Msg2	u_prev1sin((u_prev1 - 10.82263863)sin(ux1 + 0.73160326) + sin(tu_prev2) + cos(u_prev1)/0.27612534)
			Node	u_prev + ((-u_prev + ux0.25640112)2 - (ux - (-0.6112448)uy))*0.0033418469

Appendix

Table 3: Unedited symbolic expressions discovered for the Convection-Diffusion Equation across varying noise levels.

Noise	MSE(Noise)	MSE(Original)	Type	Equation
0.0%	/	9.6899E-04	Msg1	-sin(t/(-6.233686) + y_prev2) - 0.6374596
			Msg2	(cos((-0.118906446 * t) - cos(x_prev1 + 0.21283843)) - y_prev2) * 0.6010129
			Node1	sin(y - 1.5701683)(ux + uy - sin(x))cos(sin(t(-0.1646834)) - 10.1230068)*0.9365822
			Node2	((sin(t * -0.13451573) + 1.6071377) * sin(y)) * ((cos(x) - 0.007157283) * -0.6139257)
0.1%	6.2724E-04	6.2718E-04	Msg1	(-x_prev2 + y_prev1)*0.015265973
			Msg2	0.15940122 + (t - x_prev2)*(-0.026457742)

Appendix

Table 4: Unedited symbolic expressions discovered for the Incompressible Navier-Stokes Equations across varying noise levels.

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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.

Abstract

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