ICLR 2026

Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs

Cheng Jing^1,*Uvini Balasuriya^1,*Abhishek Verma² Kallol Bera² Shahid Rauf² Kookjin Lee^1,†

* Equal Contribution† Corresponding Author: [email protected]

¹Arizona State University²Applied Materials

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Uvini is seeking Summer 2026 internship opportunities —Uvini's CV

Plasma simulation and Gray-Scott reaction-diffusion patterns

Left: 1D capacitively coupled plasma (CCP) — electron density and electric field dynamics. Right: Gray–Scott reaction-diffusion — labyrinthine pattern formation.

Abstract

Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray–Scott system, show that our methods achieve up to 55∼72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.

55–72%

Error reduction over baselines

hpFNOₓ

Best model on both benchmarks

Principled architectural extensions

1D CCP

New plasma benchmark dataset

Method

Two principled extensions to the standard Fourier Neural Operator (FNO) architecture.

hpFNO — Hypernetwork-based Parameter Modulation

For parameterized PDEs: conditioning the operator on physical parameters at every Fourier layer.

v_{\ell+1}(x) = \sigma\!\left( W v_\ell(x) + \bigl(\mathcal{K}(a;\varphi)\,v_\ell\bigr)(x) + \mathbf{s}_\ell(x,\mu) \right)

A compact hypernetwork $f_{\mathrm{hyper}}$ takes physical parameters $\mu$ and position $x$ , producing layer-specific shift terms $\mathbf{s}_\ell(x,\mu)$ . Unlike HyperFNO which infers all core weights, hpFNO only infers shift biases — adding parameter-dependent modulation with negligible overhead.

FNOₓ — Spectral Coupling for Multi-variable PDEs

For coupled systems: cross-variable interaction entirely in Fourier space.

\tilde{Z}' = f_{\mathrm{enc}}\!\left(\mathcal{F}v^\alpha,\, \mathcal{F}v^\beta\right), \quad \tilde{Z}'' = R_\varphi \cdot \tilde{Z}', \quad (\tilde{v}^\alpha,\tilde{v}^\beta) = f_{\mathrm{dec}}(\tilde{Z}'')

Variables $v^\alpha$ and $v^\beta$ are independently transformed to Fourier space, jointly encoded by $f_{\mathrm{enc}}$ into a shared latent, filtered by a single spectral kernel $R_\varphi$ , then decoded by $f_{\mathrm{dec}}$ back into per-variable representations. This captures cross-variable interactions while preserving FNO's efficiency.

Combined FNOx + hpFNO architecture diagram

Combined architecture: two coupled variables ( $\alpha$ , $\beta$ ) pass through shared spectral coupling ( $f_{\mathrm{enc}} \to R_\varphi \to f_{\mathrm{dec}}$ ) while a HyperNetwork injects parameter-conditioned shifts $s(\mu,x)$ at every layer.

Results

Evaluated on two benchmark PDEs. Performance measured in normalized RMSE (nRMSE, lower is better). All experiments repeated 5× with different random seeds.

1D Capacitively Coupled Plasma (CCP)

Novel benchmark introduced in this work. Models are tested on three physical parameters independently: reaction rate $R_0$ , driving voltage $V_0$ , and ion mass $m_i$ . Reported as mean ± std.

Model	$R_0$	$V_0$	$m_i$
FNOm	0.0403	0.0791	0.0363
FNOc	0.0375	0.0873	0.0299
CFNO	0.0315	0.0428	0.0333
HyperFNOc	0.0278	0.0355	0.0253
MWTc	0.0409	0.0639	0.0403
CMWNO	0.0312	0.0526	0.0241
DONc	0.0844	0.2147	0.1035
U-NETc	0.1084	0.0844	0.1719
FNOx	0.0193	0.0345	0.0212
pFNOx	0.0194	0.0278	0.0142
hpFNOx ★	0.0154	0.0192	0.0128

nRMSE mean (±std omitted for space). Source: Table 1 — Jing et al., ICLR 2026

Governing Equations

\partial_t n_e = -\partial_x \Gamma_e + R

\partial_{xx} \phi = -\tfrac{e}{\epsilon_0}(n_e - n_{i0})

Parameters: $V_0 \in [100, 300]$ , $R_0 \in [2.7{\times}10^{19},\,2.7{\times}10^{20}]$ , $m_i \in [1.67{\times}10^{-26},\,6.68{\times}10^{-26}]$ . 100 trajectories, 9:1 train/test split.

hpFNOₓ achieves best nRMSE on all three parameters

e.g. $V_0$ : 0.0192 vs. best baseline CFNO: 0.0428 — 55% lower error

Robustness to Input History Length $T_{\mathrm{in}}$

Performance on the reaction rate case as the number of input time steps decreases. Parameterized variants (hp) remain stable even with minimal history ( $T_{\mathrm{in}}=1$ ), while non-parameterized models diverge (†).

Model	$T_{\mathrm{in}}=10$	$T_{\mathrm{in}}=5$	$T_{\mathrm{in}}=2$	$T_{\mathrm{in}}=1$
FNOc	0.0375 ±0.0055	0.1324 ±0.0304	1.0048†	1.4484†
pFNOc	0.0334 ±0.0101	0.0923 ±0.0276	0.2151‡ ±0.0374	0.3757‡ ±0.0679
hpFNOc	0.0196 ±0.0021	0.0804 ±0.0237	0.1515‡ ±0.0070	0.1609‡ ±0.0043
FNOx	0.0193 ±0.0059	0.0406 ±0.0093	1.2832†	1.8143†
pFNOx	0.0194 ±0.0075	0.0464 ±0.0158	0.1640‡ ±0.0155	0.2522‡ ±0.1376
hpFNOx ★	0.0154 ±0.0029	0.0317 ±0.0022	0.1324‡ ±0.0242	0.1372‡ ±0.0100

† Model fails to learn the dynamics. ‡ Averaged from best 5 of 10 runs due to training instability. Source: Table 2 — Jing et al., ICLR 2026

Efficiency Analysis

hpFNOₓ achieves the best accuracy at comparable model size and training time to standard FNO — a Pareto-optimal solution. Results from the CCP varying driving voltage ( $V_0$ ) scenario. Source: Figure 4, Jing et al., ICLR 2026.

FNO Baselines

HyperFNO

Wavelet-based

Other NOs

Ours

(a) Model Size vs. Accuracy

Our methods cluster at low parameter count with best accuracy

(b) Training Time vs. Accuracy

Our methods achieve best accuracy without sacrificing training speed

Citation

If you find this work useful, please cite:

@inproceedings{jingextending,
  title={Extending Fourier Neural Operators for Modeling
         Parameterized and Coupled PDEs},
  author={Jing, Cheng and Mudiyanselage, Uvini Balasuriya
          and Verma, Abhishek and Bera, Kallol
          and Rauf, Shahid and Lee, Kookjin},
  booktitle={The Fourteenth International Conference
             on Learning Representations}
}