Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

Jiaquan Zhang; Fachrina Dewi Puspitasari; Songbo Zhang; Yibei Liu; Kuien Liu; Caiyan Qin; Fan Mo; Peng Wang

cs.LGFeb 18, 2026

Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

Authors: Jiaquan Zhang, Fachrina Dewi Puspitasari, Songbo Zhang, Yibei Liu, Kuien Liu, Caiyan Qin, Fan Mo, Peng Wang,

Organizations: School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, China · Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China · Institute of Software Chinese Academy of Sciences, Beijing, China · School of Robotics and Advanced Manufacture, Harbin Institute of Technology, Shenzhen, China ·

Abstract

Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient plug-and-play module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50% at the cost of 2.26% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.

Figures & tables

Figure 1: Conceptual comparison between common residual updates in existing neural operators vs. our geometric constrained updates. Our method, termed MCL, addresses the challenge of physical law violation in neural operator update by enforcing geometric inductive bias through Lie algebra parameterization.

Figure 2: Framework of our method, termed MCL, when integrated in a neural operator. MCL replaces the standard unconstrained Euclidean space-based update with geometric-constrained updates based on Lie algebra group action. MCL serves as an efficient plug-and-play module for existing neural operators.

Equation	Param.	Step	Model	Model only			Model + MCL
	Param.	Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
1-D Burgers	0.001	1	FNO	0.000550	0.046626	0.068925	0.000327	0.035689	0.049867
			U-NO	0.042886	0.398778	0.415302	0.039507	0.350689	0.367253
			GINO	0.003950	0.147548	0.192726	0.003891	0.146621	0.192288
		10	FNO	0.000214	0.037893	0.050876	0.000108	0.026429	0.034554
			U-NO	0.025308	0.412722	0.417508	0.020423	0.398555	0.404843

Table 1: Evaluation on MCL integration to neural operator models under various equations, parameters, and rollout steps. Best results are in bold .

Figure 3: Prediction error plot of PDEs on FNO (left) and FNO+MCL (right) against true solution.

Figure 4: Evolution of energy (left) and entropy (right) on 1-D Burgers equation with parameter values 0.001 (top), 0.01 (middle), and 0.1 (bottom).

Param.	Step	Model	MSE	Rel_L2	Rel_H1
0.1	1	FNO +MCL	0.000016	0.004433	0.005201
	1	FNO +MLP	0.000019	0.004549	0.005406
	25	FNO +MCL	0.000031	0.006246	0.006954
	25	FNO +MLP	0.000034	0.008008	0.008762
1.0	1	FNO +MCL	0.000064	0.008839	0.009975
	1	FNO +MLP	0.000078	0.009654	0.010835

Table 2: Ablation on MCL contribution on 1-D Advection. Best results are in bold .

Model	# of param.	# of param+MCL	% param. add
FNO	23,937	25,221	5.36
U-NO	2,708,673	2,732,234	0.87
GINO	442,194	444,567	0.54

Table 3: Total parameter count of original model vs. integration with MCL.

Rank dim.	Step	MSE	Rel_L2	Rel_H1
3	1	0.000021	0.000020	0.000017
	10	0.000003	0.004529	0.004668
	25	0.000001	0.003394	0.003454
	30	0.000001	0.003382	0.003436
5	1	0.000020	0.008317	0.008839
	10	0.000003	0.004339	0.004482

Table 4: Sensitivity test on hyperparameter.

Step	Model	Model only			Model + MCL
Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
1	FNO	0.094958	0.006878	0.007343	0.003067	0.005678	0.006499
	U-NO	0.121584	0.090440	0.099027	0.117830	0.085402	0.094537
	GINO	0.014434	0.013638	0.021241	0.013268	0.013185	0.020583
3	FNO	0.002689	0.006452	0.007013	0.002607	0.005090	0.005856
	U-NO	0.073323	0.040074	0.047294	0.044330	0.037223	0.046833

Table A1 : 2-D Navier-Stokes

Parameter	Model	Model only			Model + MCL
Parameter	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
0.01	FNO	0.00057275	2.59687380	2.61742788	0.00019451	1.04014158	1.06347932
	U-NO	0.00022823	1.69248305	1.78475725	0.00021132	1.36236886	1.64780834
	GINO	0.00106853	5.07440402	5.05898289	0.00106853	5.06830841	5.05290873
0.1	FNO	0.00044092	0.44783878	0.45378846	0.00021056	0.24423277	0.25018481
	U-NO	0.00022993	0.33987113	0.34327716	0.00023840	0.15001997	0.16205981

Table A2 : 2-D Darcy Flow

Parameter	Step	Model	Model only			Model + MCL
Parameter	Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
0.1	1	FNO	0.000010	0.006182	0.006210	0.000001	0.002284	0.002361
		U-NO	0.033765	0.452617	0.476068	0.002880	0.099611	0.104958
		GINO	0.000219	0.039598	0.043709	0.000198	0.038372	0.042604
	10	FNO	0.000002	0.003629	0.003636	0.000001	0.001334	0.001347
		U-NO	0.005832	0.205232	0.225318	0.001769	0.067778	0.070619

Table A3 : 1-D Burgers

Parameter	Step	Model	Model only			Model + MCL
Parameter	Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
0.1	1	FNO	0.000037	0.005690	0.006920	0.000016	0.004433	0.005201
		U-NO	0.002880	0.099611	0.104958	0.002003	0.069950	0.079686
		GINO	0.002394	0.060020	0.069219	0.001251	0.038808	0.042497
	10	FNO	0.000096	0.009490	0.010656	0.000021	0.004975	0.005685
		U-NO	0.005685	0.07778	0.070619	0.001110	0.052989	0.059960

Table A4: 1-D Advection

Step	Model	Model only			Model + MCL
Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
1	FNO	0.000013	0.013330	0.013673	0.000000	0.001321	0.002699
	U-NO	0.016053	0.444866	0.447889	0.005528	0.264650	0.269736
	GINO	0.269736	0.024395	0.025905	0.000138	0.043518	0.048344
10	FNO	0.000001	0.002817	0.003851	0.000000	0.001211	0.002637
	U-NO	0.007221	0.275725	0.276714	0.004055	0.190123	0.190769

Table A5 : 1-D Diffusion

Step	Model	Model only			Model + MCL
Step	Model	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
1	FNO	0.000021	0.004370	0.008327	0.000010	0.003084	0.005210
	U-NO	0.035991	0.172072	0.172072	0.012362	0.101253	0.112789
	GINO	0.020287	0.131313	0.149931	0.020275	0.131233	0.150866
10	FNO	0.000015	0.003681	0.005530	0.000004	0.002056	0.003120
	U-NO	0.021521	0.137780	0.141981	0.016664	0.117650	0.122598

Table A6 : 2-D Shallow Water

Parameter	Step	FNO + MCL			FNO + MLP
Parameter	Step	MSE	Rel_L2	Rel_H1	MSE	Rel_L2	Rel_H1
0.1	1	0.000016	0.004433	0.005201	0.000019	0.004549	0.005406
	10	0.000021	0.004890	0.005646	0.000023	0.010975	0.010685
	25	0.000031	0.006246	0.006954	0.000034	0.008008	0.008762
	30	0.000037	0.006938	0.007660	0.000039	0.013526	0.014306
0.4	1	0.000044	0.007285	0.008363	0.000070	0.009049	0.010143

Table A7 : Ablation on 1-D Advection

Figure A1 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 1-D linear advection equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Figure A2 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 1-D Burgers equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Figure A3 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 1-D diffusion-sorption equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Figure A4 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 2-D Darcy flow equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Figure A5 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 2-D shallow-water equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Figure A6 : Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 2-D Navier-Stokes equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

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Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.

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