Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

Xiaoyang Xie; Clarence W. Rowley

math.NAAug 24, 2026

Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

Authors: Xiaoyang Xie, Clarence W. Rowley

Organizations: Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, 08540, New Jersey, United States. · Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, 08540, New Jersey, United States.

Abstract

In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

Figures & tables

Figure 1: Schematic of a trajectory in the full space and its tracking trajectory on the inertial manifold.

$t_{\mathrm{end}}=0.5$
	FNO	RNO	IMNO-SE( $\alpha=0.1$ )	IMNO-SE( $\alpha=0$ )
Params	418433	623533	538314	538314
$\nu=0.002$	$2.91\pm 0.07$ %	$3.55\pm 0.05$ %	$2.25\pm 0.03$ %	$1.94\pm 0.04$ %
$\nu=0.01$	$1.35\pm 0.08$ %	$1.79\pm 0.03$ %	$1.16\pm 0.03$ %	$0.78\pm 0.03$ %
$t_{\mathrm{end}}=2$
$\nu=0.002$	$53.3\pm 43.8$ $\%^{*}$	$4.12\pm 0.92$ %	$2.45\pm 0.12$ %	$1.95\pm 0.12$ %

Table 1: Relative

L^{2}

error of FNO, RNO and IMNO-SE for the Burgers Equation

Panel of Figure 3 — Figure 3: IMNO-SE ( $\alpha=0.1$ ) predictions for 1D Burgers equation ( $\nu=0.01$ ).

$t_{\mathrm{end}}=0.5$
	FNO	RNO	IMNO( $\alpha=0.1$ )	IMNO( $\alpha=0$ )
Params	418497	623617	491377	491377
$f=0$	$0.59\pm 0.09$ %	$0.89\pm 0.05$ %	$0.51\pm 0.004$ %	$0.43\pm 0.02$ %
$f\neq 0$	$0.77\pm 0.06$ %	$1.10\pm 0.03$ %	$0.54\pm 0.004$ %	$0.48\pm 0.03$ %
$t_{\mathrm{end}}=2$
$f=0$	$16.60\pm 20.34$ $\%^{*}$	$1.14\pm 0.03$ %	$0.58\pm 0.0003$ %	$0.41\pm 0.006$ %

Table 2: Relative

L^{2}

error of FNO, RNO and IMNO for the nonlocal Burgers equation

Panel of Figure 4 — Figure 4: IMNO ( $\alpha=0.1$ ) predictions for the nonlocal Burgers equation.

Panel of Figure 5 — Figure 5: Representative hidden-state trajectories projected onto the top three PCA modes for the nonlocal Burgers equation. (Dots indicate the initial states of different trajectories, while crosses mark their corresponding final states at $t=2$ .)

$t_{\mathrm{end}}=20$
	FNO	RNO	IMNO-SE( $\alpha=0.1$ )	IMNO-SE( $\alpha=0$ )
Params	418433	623533	538314	538314
$L=4\pi$	$1.04\pm 0.20$ %	$0.42\pm 0.01$ %	$0.52\pm 0.02$ %	$0.46\pm 0.003$ %
$L=8\pi$	$1.74\pm 0.05$ %	$1.19\pm 0.01$ %	$4.25\pm 0.17$ %	$2.99\pm 0.42$ %
$t_{\mathrm{end}}=40$
$L=4\pi$	$87.67\pm 11.85$ $\%^{*}$	$2.19\pm 0.23$ %	$0.89\pm 0.05$ %	$0.84\pm 0.02$ %

Table 3: Relative

L^{2}

error of FNO, RNO and IMNO-SE for the KS equation

Panel of Figure 6 — Figure 6: IMNO-SE ( $\alpha=0.1$ ) predictions for the KS equation at $L=4\pi$

Figure 7: Representative hidden-state trajectories projected onto the top three PCA modes for the KS equation at $L=4\pi$ .

	FNO	RNO	IMNO-SE( $\alpha=1$ )	IMNO-SE( $\alpha=0.1$ )	IMNO-SE( $\alpha=0$ )
Params	418819	623939	619340	619340	619340
$t_{\mathrm{end}}=0.5$	$12.05\pm 13.44$ $\%^{*}$	$3.24\pm 0.06$ %	$5.12\pm 0.05$ %	$3.54\pm 0.29$ %	$2.69\pm 0.10$ %
$t_{\mathrm{end}}=1.0$	$84.47\pm 5.14$ $\%^{*}$	$4.79\pm 0.64$ %	$4.49\pm 0.41$ %	$4.33\pm 0.15$ %	$3.53\pm 0.17$ %

Table 4: Relative

L^{2}

error of FNO, RNO, and IMNO-SE for the 1D NS Equation

Panel of Figure 8 — Figure 8: IMNO-SE ( $\alpha=0.1,1$ ) predictions for the 1D NS equation.

	FNO	RNO	IMNO-SE( $\alpha=0.1$ )	IMNO-SE( $\alpha=0$ )
Params	465784	697084	616294	616294
$t_{\mathrm{end}}=0.5$	$25.64\pm 1.22$ %	$20.71\pm 0.12$ %	$10.46\pm 0.67$ %	$10.48\pm 0.39$ %
$t_{\mathrm{end}}=1.0$	$52.46\pm 0.99$ $\%^{*}$	$24.73\pm 1.58$ %	$14.73\pm 1.15$ %	$14.68\pm 1.43$ %

Table 5: Relative

L^{2}

error of FNO, RNO and IMNO-SE for the 2D NS equation

Panel of Figure 9 — Figure 9: Comparison of FNO, RNO and IMNO-SE ( $\alpha=0.1$ ) for the 2D compressible NS equation at $t=0.05,0.25,0.50$ , and $1.00$ .

	FNO	RNO	IMNO-SE( $\alpha=0.1$ )	IMNO-SE( $\alpha=0$ )
Params	465357	696657	578146	578146
$t_{\mathrm{end}}=10$	$3.43\pm 0.08$ %	$6.13\pm 0.11$ %	$3.76\pm 0.12$ %	$2.96\pm 0.02$ %
$t_{\mathrm{end}}=20$	$3.10\pm 0.11$ %	$5.16\pm 0.06$ %	$3.94\pm 0.60$ %	$2.75\pm 0.11$ %

Table 6: Relative

L^{2}

error of FNO, RNO and IMNO-SE for the 2D incompressible NS equation

Panel of Figure 10 — Figure 10: IMNO-SE ( $\alpha=0.1$ ) predictions for the incompressible 2D NS equation at $t=1,2,5,10,20$ .

Panel of Figure 11 — Figure 11: IMNO-SE ( $\alpha=0.1$ ) predictions for the incompressible 2D Navier–Stokes equation at $t=1,2,5,10,20$ in the turbulence regime $\nu=10^{-3}$ .

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Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

Abstract

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