Learning Physical Operators using Neural Operators

Vignesh Gopakumar; Ander Gray; Dan Giles; Lorenzo Zanisi; Matt J. Kusner; Timo Betcke; Stanislas Pamela; Marc Peter Deisenroth

cs.LGFeb 26, 2026

Learning Physical Operators using Neural Operators

Authors: Vignesh Gopakumar, Ander Gray, Dan Giles, Lorenzo Zanisi, Matt J. Kusner, Timo Betcke, Stanislas Pamela, Marc Peter Deisenroth

Organizations: UCL Centre for Artificial Intelligence · UK Atomic Energy Authority · LIX, CNRS, ´Ecole Polytechnique · Polytechnique Montr´eal · Mila - Quebec AI Institute

Abstract

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier--Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Figures & tables

Figure 1: Comparison of operator learning approaches for PDE solving. The figure contrasts three methods: (top left) the traditional autoregressive approach where a single neural operator learns the solution operator mapping directly from state $u^{n}$ to $u^{n+1}$ ; (bottom left) the neural ODE approach where a neural operator learns the dynamics operator $\derivative{u}{t}$ integrated with an ODE solver; and (right) the proposed OpsSplit method where individual neural operators learn specific physical operators $(\nabla\crossproduct,\nabla\cdot)$ and are combined via operator splitting to compute $\derivative{u}{t}$ which is integrated using an ODE solver. The OpsSplit approach decomposes the PDE into its constituent physical operators, with linear operators approximated by convolutions and non-linear operators learned by neural networks, enabling physics-informed and modular PDE solving.

					Train Time	NRMSE
Operator Learning	Model	Time Stepping	OpsSplit	Parameters	(hrs:mins)	Test	t-extrapolate	OOD	OOD+t-extrapolate
Solution	FNO	Autoregressive	False	13437314	1:03	0.2770 $\pm$ 0.0491	0.6860 ± 0.26476	0.4821 ± 0.0639	0.8986 ± 0.2727
Dynamics	FNO	Euler	False	13437314	1:04	0.0325 $\pm$ 0.0165	0.0536 ± 0.0246	0.1276 ± 0.0081	0.3038 ± 0.0119
Physical	FNO	Euler	True	13437314	1:06	0.0297 $\pm$ 0.0017	0.0630 ± 0.0111	0.1160 ± 0.0162	0.2185 ± 0.0846
Solution	U-Net	Autoregressive	False	31384322	1:22	0.9508 ± 0.1369	1.3183 ± 0.1318	0.8192 ± 0.2007	1.1872 ± 0.2534
Dynamics	U-Net	Euler	False	31384322	1:22	0.1006 ± 0.0133	0.2212 ± 0.0315	0.2823 ± 0.0154	0.5135 ± 0.0331

Table 1: Incompressible Navier–Stokes: Performance comparison across methods and model architectures. The models and methods are set up to have comparable parameter sizes while exploring a certain architecture. Best performance within each test setting is given in bold. Our method of using neural operators to learn physical operators offers the best performance across most architectures. OOD refers to a different parameterisation of the initial condition and viscosity than that used for training ( appendix J ).

Panel of Figure 2 — Figure 2: Rollout error for incompressible Navier–Stokes equations. FNO ( figs. 2(a) and 2(b) ) and CNO ( figs. 2(c) and 2(d) ) predictive error growth for in-distribution and out-of-distribution cases. The temporal extrapolation region is shaded orange. OpsSplit accumulates fewer errors and provides more stable temporal rollout than autoregressive and neural ODE methods across architectures and scenarios.

Panel of Figure 3 — Figure 3: Neural operator learned convection ( fig. 3(a) ) versus numerical method. NO captures convection nuances and advection-driven flow trends. Qualitative interpretability study: learned convection exists in latent space; numerical convection shown in physical space, normalised to [-1, 1].

Figure 4: Continuity equation ( eq. 14 ) violation across FNO predictions for OOD modeling.

					Train Time	NRMSE
Operator Learning	Model	Time Stepping	OpsSplit	Parameters	(hrs:mins)	Test	t-extrapolate	OOD	OOD+t-extrapolate
Solution	FNO	Autoregressive	False	13437828	1:53	0.0938 $\pm$ 0.0131	0.2462 $\pm$ 0.0443	0.1209 $\pm$ 0.0133	0.2792 $\pm$ 0.0530
Dynamics	FNO	Euler	False	13437828	1:56	0.0245 $\pm$ 0.0039	0.0854 $\pm$ 0.0102	0.0763 $\pm$ 0.0130	0.1185 $\pm$ 0.0154
Physical	FNO	Euler	True	13454787	3:08	0.0602 $\pm$ 0.0090	0.0959 $\pm$ 0.0182	0.0573 $\pm$ 0.0063	0.0751 $\pm$ 0.0105
Solution	U-Net	Autoregressive	False	31385604	1:53	0.5963 $\pm$ 0.0716	1.1914 $\pm$ 0.2025	0.6069 $\pm$ 0.1153	1.2384 $\pm$ 0.1610
Dynamics	U-Net	Euler	False	31385604	1:54	0.1222 $\pm$ 0.0196	0.2912 $\pm$ 0.0320	0.1477 $\pm$ 0.0222	0.2989 $\pm$ 0.0538

Table 2: Compressible Navier–Stokes: Performance comparison across methods and model architectures. The models and methods are set up to have comparable parameter sizes while exploring a certain architecture. Best performance within each test setting is given in bold. Our method of using neural operators to learn physical operators offers the best performance across most architectures. OOD refers to a different parameterisation of the initial condition and adiabatic index than that used for training ( appendix K ). The higher computational times for OpsSplit is observed due to the several forward passes required to estimate the physical operators.

Panel of Figure 5 — Figure 5: Rollout error for the compressible Navier–Stokes equations. Similar to fig. 2 , show the rollout error of various methods for an FNO and U-Net for both in and out-of-distribution scenarios. In most cases, across neural operator architectures, our method of deploying operator splitting to learn the physical operator accumulates less error and provides a more stable temporal rollout than autoregressive and neural ODE-based methods.

					Train Time	NRMSE
Operator Learning	Model	Time Stepping	OpsSplit	Parameters	(hrs:mins)	In-Distribution	Out-of-Distribution
Solution	GNO	Autoregressive	False	330,370	6:25	$0.4131\pm 0.0475$	$0.5697\pm 0.0530$
Dynamics	GNO	Euler	False	330,370	6:26	$0.3276\pm 0.0398$	$0.4510\pm 0.0571$
Physical	GNO	Euler	True	660,740	9:31	$\mathbf{0.2963\pm 0.0043}$	$\mathbf{0.3621\pm 0.0510}$

Table 3: Incompressible Navier–Stokes on irregular geometries: Performance comparison across methods and model architectures. Best performance within each test setting is shown in bold. Our method of using neural operators to learn physical operators achieves the best performance. In-distribution refers to modelling within the training regime, whereas out-of-distribution refers to extrapolation further in time on a different validation dataset. The higher computational time for OpsSplit is due to the multiple models required for each physical operator.

Figure 6: Loss convergence for OpsSplit models on incompressible Navier–Stokes. The pre-trained model learns all operators from scratch. The fine-tuned model initialises the convection operator with trained weights from the compressible case, showing quicker convergence. For more details, refer appendix I .

Appendix figures & tables46 assets

Supplementary material from the paper’s appendix.

Appendix

Work (citation)	Physics-informed	Continuous-in-time	Modular/ Mixture of Experts	Interpretable	Extrapolation / Generalisation
[ Li et al., 2021 ] (FNO)	✗	✗	✗	✗	✗
[ Lu et al., 2021 ] (DeepONet)	✗	✓	✓	✗	✗
[ Raissi et al., 2019 ] (PINNs)	✓	✓	✗	✓	✓
[ Rackauckas et al., 2021 ] (UDE)	✓	✓	✓	✗	✗
[ Li et al., 2024 ] (PINO)	✓	✗	✗	✗	✓
[ Koch et al., 2025 ]	✓	✓	✓	✗	✓

Appendix

Table 4: Comparison of different approaches for Neural PDE solvers. ✓: Yes, ✗ : No, ✓ : Partial.

Panel of Figure 7 — Figure 7: Incompressible Navier–Stokes: comparing across convection operator obtained numerically against those learnt using various neural operators.

Panel of Figure 8 — Figure 8: Compressible Navier–Stokes: comparing across convection and vector divergence operator obtained numerically against those learnt using various neural operators.

Panel of Figure 9 — Figure 9: Incompressible Navier–Stokes: Rollout Error of the the various OpsSplit ablations as described in table 5 . We experiment each ablation across both in and out-of-distribution. While we notice that higher order finite-difference stencils deployed to model the linear operators perform best within distribution, they fail to extend this to outside distribution. Learnt approximations of the linear diffusion operator either using a linear convolutional operator (5) or a neural operator (6) allows for better generalisation.

Ablation	Convection	Diffusion		Train Time	NRMSE
	$(\mathbf{v}\cdot\nabla)\mathbf{v}$	$\nabla^{2}\mathbf{v}$	Parameters	(hrs:mins)	Test	t-extrapolate	OOD	OOD+t-extrapolate
1	NO	FD (2)	13437321	1:11	0.0270	0.0796	0.1471	0.4731
2	NO	FD (4)	13437339	1:12	0.0244	0.0600	0.1993	0.6995
3	NO	FD (6)	13437350	1:13	0.0221	0.0504	0.1576	0.5303
4	NO	FD (8)	13437378	1:13	0.0214	0.0568	0.4195	1.5992
5	NO	Linear	13438196	1:13	0.0283	0.0823	0.1581	0.3693

Appendix

Table 5: Incompressible Navier–Stokes: Performance comparison across different methods of operator splitting and subsequent operator choices.

FD(n)

represents a fixed finite difference kernel of order

n

, Linear corresponds to a learnable linear convolution, and

NO

represents an FNO.

Panel of Figure 10 — Figure 10: Compressible Navier–Stokes: Rollout Error of the the various OpsSplit ablations as described in table 6 . We experiment each ablation across both in and out-of-distribution. We notice that when we approximate nonlinear operators using FD stencils the prediction blows up when extrapolated in time. The general trend we notice is that approximating using neural operators offer better stability both in and out-of-distribution.

Ablation	Continuity	Convection	Pressure	Pressure	Pressure		Train Time	NRMSE
			Gradient	Advection	Expansion
	$\nabla\cdot(\rho\mathbf{v})$	$(\mathbf{v}\cdot\nabla)\mathbf{v}$	$\nabla\mathbf{P}$	$\mathbf{v}\cdot\nabla P$	$P(\nabla\cdot\mathbf{v})$	Parameters	(hrs:mins)	Test	t-extrapolate	OOD	OOD+t-extrapolate
1	NO 1	NO 2	FD (4)	FD (4)	FD (4)	26874627	3:56	0.0824	-	0.0851	-
2	NO 1	NO 2	FD (4)	FD (4)	NO 3	40311940	5:43	0.0762	0.1441	0.0785	0.1564
3	NO 1	NO 2	Linear	Linear	Linear	26874666	3:52	0.0639	0.1115	0.2061	0.4201
4	NO 1	NO 2	Linear	Linear	NO 3	40311979	5:41	0.0866	0.1177	0.1329	0.2839

Appendix

Table 6: Compressible Navier–Stokes: Performance comparison across different methods of operator splitting and subsequent operator choices.

FD(n)

represents a fixed finite difference kernel of order

n

, Linear corresponds to a learnable linear convolution, and

NO

represents an FNO.

NO_{i}

represents the

i^{th}

neural operator within the system.

Panel of Figure 11 — Figure 11: Incompressible Navier–Stokes: Rollout length of the FNO deployed across each method for both in and out-of-distribution.

Panel of Figure 12 — Figure 12: Incompressible Navier–Stokes: Data Efficiency of the FNO deployed across each method for both in and out-of-distribution.

Panel of Figure 13 — Figure 13: Incompressible Navier–Stokes: Data Efficiency of the FNO deployed across each method for both in and out-of-distribution.

Panel of Figure 14 — Figure 14: Incompressible Navier–Stokes: Impact of finesse of ODE solver used for temporal integration of the FNO deployed across each method for both in and out-of-distribution.

Panel of Figure 15 — Figure 15: Incompressible Navier–Stokes: Train and test loss convergences across training methods for the FNO

Panel of Figure 16 — Figure 16: Compressible Navier–Stokes: Train and test loss convergences across training methods for the FNO

Panel of Figure 17 — Figure 17: Incompressible Navier–Stokes: Comparison of training and test loss convergence for FNO-based OpsSplit models. The pre-trained baseline learns all operators from scratch with random initialisation. The fine-tuned model initialises the convection operator using weights pre-trained on the compressible Navier–Stokes equations ( eq. 21 ), demonstrating accelerated convergence through cross-PDE transfer learning.

Panel of Figure 18 — Figure 18: Compressible Navier–Stokes: Comparison of training and test loss convergence for FNO-based OpsSplit models. The pre-trained baseline learns all operators from scratch with random initialisation. The fine-tuned model initialises the convection operator using weights pre-trained on the incompressible Navier–Stokes equations ( eq. 16 ), demonstrating that transfer learning benefits are bidirectional across fluid dynamics regimes.

Parameter	Training	OOD Testing	Type
Velocity x-axis $(u_{0})$	$[0.5,1.0]$	$[0.1,0.5]$	Continuous
Velocity y-axis $(v_{0})$	$[0.5,1.0]$	$[0.1,0.5]$	Continuous
viscosity $\nu$	0.001	0.01	Discrete

Appendix

Table 7: Parameterisations of the 2D incompressible Navier–Stokes equations utilised for training and OOD testing

Model	Configuration Details - AR, NODE, OpsSplit
FNO	in_channels: 2 out_channels: 2 modes: 32 width: 64 n_layers: 6 activation_func: GeLU
U-Net	in_channels: 2 out_channels: 2 initial_width: 64 activation_func: Tanh
ViT	patch_size: 4 embed_dim: 512 in_channels: 2 out_channels: 2 time_channels: 1 depth: 12 num_heads: 10 activation_func: GeLU
CNO	Nx: 100 N_layers: 4 N_res: 4 N_res_neck: 2 in_channels: 2 out_channels: 2 channel_multiplier: 16 activation_func: LReLU
UNO	arch: UNO in_channels: 2 out_channels: 2 width: 64 projection_channels: 256 n_layers: 5 uno_out_channels: [32,64,64,64,32] uno_n_modes: [[16,16],[8,8],[8,8],[8,8],[16,16]] domain_padding: 0.2 norm: group_norm activation_func: GeLU

Appendix

Table 8: Model configuration details of neural operators used for modelling the incompressible Navier–Stokes equations. All three methods - AR, NODE, and OpsSplit utilised NO models with the same configuration as mentioned above.

Panel of Figure 19 — Figure 19: Incompressible Navier–Stokes: Model prediction within test distribution for Autoregressive FNO

Panel of Figure 20 — Figure 20: Incompressible Navier–Stokes: Model prediction within test distribution for Neural-ODE FNO

Panel of Figure 21 — Figure 21: Incompressible Navier–Stokes: Model prediction within test distribution for OpsSplit FNO

Panel of Figure 22 — Figure 22: Incompressible Navier–Stokes: Model prediction within test distribution for Autoregressive U-Net

Panel of Figure 23 — Figure 23: Incompressible Navier–Stokes: Model prediction within test distribution for Neural-ODE U-Net

Panel of Figure 24 — Figure 24: Incompressible Navier–Stokes: Model prediction within test distribution for OpsSplit U-Net

Panel of Figure 25 — Figure 25: Incompressible Navier–Stokes: Model prediction within test distribution for Autoregressive ViT

Panel of Figure 26 — Figure 26: Incompressible Navier–Stokes: Model prediction within test distribution for Neural-ODE ViT

Panel of Figure 27 — Figure 27: Incompressible Navier–Stokes: Model prediction within test distribution for OpsSplit ViT

Panel of Figure 28 — Figure 28: Incompressible Navier–Stokes: Model prediction within test distribution for Autoregressive CNO

Panel of Figure 29 — Figure 29: Incompressible Navier–Stokes: Model prediction within test distribution for Neural-ODE CNO

Panel of Figure 30 — Figure 30: Incompressible Navier–Stokes: Model prediction within test distribution for OpsSplit CNO

Panel of Figure 31 — Figure 31: Incompressible Navier–Stokes: Model prediction within test distribution for Autoregressive UNO

Panel of Figure 32 — Figure 32: Incompressible Navier–Stokes: Model prediction within test distribution for Neural-ODE UNO

Panel of Figure 33 — Figure 33: Incompressible Navier–Stokes: Model prediction within test distribution for OpsSplit UNO

Parameter	Training	OOD Testing	Type
$\alpha$	$[0.1,0.5]$	$[0.5,1.0]$	Continuous
$\beta$	$[1.0,5.0]$	$[5.0,10.0]$	Continuous
$\gamma$	$\frac{5}{3}$	$\frac{2}{3}$	Discrete

Appendix

Table 9: Parameterisation of the 2D Euler Fluid equations utilised for training and OOD testing

Model	Configuration Details - AR, NODE	OpsSplit
FNO	in_channels: 4 out_channels: 4 modes: 32 width: 64 n_layers: 6 activation_func: GeLU	in_channels: 2 (conv), 3 (div) out_channels: 2 (conv), 1 (div) modes: 32 width: 64 n_layers: 3 activation_func: GeLU
U-Net	in_channels: 4 out_channels: 4 initial_width: 64 activation_func: Tanh	in_channels: 2 (conv), 3 (div) out_channels: 2 (conv), 1 (div) initial_width: 32 activation_func: Tanh
ViT	patch_size: 4 embed_dim: 512 in_channels: 4 out_channels: 4 time_channels: 1 depth: 12 num_heads: 10 activation_func: GeLU	patch_size: 4 embed_dim: 256 in_channels: 2 (conv), 3 (div) out_channels: 2 (conv), 1 (div) time_channels: 1 depth: 8 num_heads: 10 activation_func: GeLU
UNO	in_channels: 4 out_channels: 4 width: 64 projection_channels: 256 n_layers: 5 uno_out_channels: [32,64,64,64,32] uno_n_modes: [[16,16],[8,8],[8,8],[8,8],[16,16]] domain_padding: 0.2 norm: group_norm activation_func: GeLU	in_channels: 2 (conv), 3 (div) out_channels: 2 (conv), 1 (div) width: 32 projection_channels: 256 n_layers: 5 uno_out_channels: [32,64,64,64,32] uno_n_modes: [[16,16],[8,8],[8,8],[8,8],[16,16]] domain_padding: 0.2 norm: group_norm activation_func: GeLU

Appendix

Table 10: Model configuration details of neural operators used for modelling the compressible Navier–Stokes equations. Both AR and NODE-based models have the same configuration, whereas the two models within OpsSplit have slightly different setups to ensure the same parameterisation levels across all methods.

Panel of Figure 34 — Figure 34: Compressible Navier–Stokes: Model prediction within test distribution for Autoregressive FNO

Panel of Figure 35 — Figure 35: Compressible Navier–Stokes: Model prediction within test distribution for Neural-ODE FNO

Panel of Figure 36 — Figure 36: Compressible Navier–Stokes: Model prediction within test distribution for OpsSplit FNO

Panel of Figure 37 — Figure 37: Compressible Navier–Stokes: Model prediction within test distribution for Autoregressive U-Net

Panel of Figure 38 — Figure 38: Compressible Navier–Stokes: Model prediction within test distribution for Neural-ODE U-Net

Panel of Figure 39 — Figure 39: Compressible Navier–Stokes: Model prediction within test distribution for OpsSplit U-Net

Panel of Figure 40 — Figure 40: Compressible Navier–Stokes: Model prediction within test distribution for Autoregressive ViT

Panel of Figure 41 — Figure 41: Compressible Navier–Stokes: Model prediction within test distribution for Neural-ODE ViT

Panel of Figure 42 — Figure 42: Compressible Navier–Stokes: Model prediction within test distribution for OpsSplit ViT

Panel of Figure 43 — Figure 43: Compressible Navier–Stokes: Model prediction within test distribution for Autoregressive UNO

Panel of Figure 44 — Figure 44: Compressible Navier–Stokes: Model prediction within test distribution for Neural-ODE UNO

Panel of Figure 45 — Figure 45: Compressible Navier–Stokes: Model prediction within test distribution for OpsSplit UNO

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