Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs
Authors: Cheng Jing
Uvini Balasuriya Mudiyanselage
Abhishek Verma
Kallol Bera
Shahid Rauf
Kookjin Lee
Venue: ICLR 2026 Poster (2026)
Keywords: neural operators
Fourier neural operators
parameterized dynamics
couple
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

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.

Published as a conference paper at ICLR 2026
EXTENDING FOURIER NEURAL OPERATORS FOR PA- RAMETERIZED AND COUPLED PDES
Cheng Jing Uvini Balasuriya' Abhishek Verma Kallol Bera² Shahid Rauf Kookjin Leek Arizona State University, AApplied Materials Inc. (cjing6, ubalasur, Kookjin Lee Lee)easu. edu,
Kallol.Bera, Shahid @amat com
ABSTRACT
Parameterized and coupled partial differential equations (PDEs) are central 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 parameter- ized dynamics, we propose a hypernetwork -bsse modulation that conditions the operator on physical parameters. For coupled systems, we conduct systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retain- ing 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 system- atic design exploration.
INTRODUCTION
Numerical simulation has long served cornerstone of scientific and engincering inquiry, under- pinning advances in areas ranging from fluid dynamics (Ferziger et al. (019)) and climate model- ing (Bauer al. 2015) material science (Rappaz al. 2003) and structural analysis (Macneal & Harder 1985). High-fidelity simulations, while indispensable, often entail substantial computa- tional cost. These costs become especially significant when conducting parameter studies, uncer- tainty quantification, and real time decision making. As a result, surrogate modeling has emerged as an essential tool to expedite simulation workflows, enabling rapid approximations that preserve first- order fidelity while dramatically reducing computational burden. Recently, deep learning techniques have gained increasing traction in this domain. Methodologies such as physics-informed informed neural net- works (PINNs) (Raissi et al.) 2019; Karniadakis et al. (2021) and neural operators (NOs) (Li et al.) 2020a by
Lu et al. 2021) offer flexible, data-driven approximations of parametrized partial differen- tial equations (PDEs), often delivering orders-of- fifccs-of-magnitueee speedups over traditional solvers while maintaining acceptable accuracy.
Despite significant progress in NO, most existing studies have considered scenarios with both vary- ing initial conditions and parameter regimes, yet they often overlook the particular, yet prevalent case of parameterized dynamics with fixed initial conditions, typified by equations of the form Ue = f (4) with u(0) = un- Such settings are ubiquitous in engineering, where model dynamics depend on physical parameters (e.g., forcing terms in inviscid Burgers' equation (Rewieńssk) 2003 Carlberg et al., (013)) diffusivitt/reaction constants in chemically reacting flows (Buffoni & Will- cox. 2010 Lee & Carlberg 2020)), and Reynolds number and viscosity in fluid dynamics (Stabile & Rozza 2018)) while initial states remain unchanged across simulations. Studying this regime is im- portant for practical applications, for example, to understand how variations in material properties, operating conditions, or control parameters affect system behavior from a fixed, well-defined state.
In this work, we focus specifically on NOs for parameterized coupled PDEs in settings. To facilitate this exploration, we build upon Fourier neural operators (FNOs) (Li et al.)
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2021). method now well-established in the community for its expressive parametric kernel in Fourier space. Our approach is designed to remain general and modular, making it adaptable to both FNO variants and other neural operator frameworks. To incorporate parameterized dynam- ies effectively, we introduce an architectural modification which the meta-physicc--knollddee encoded in the governing equations is captured by larger set of shared base parameters, while parameter-specific variations are represented through smaller set of task-dependens model param- eters (with task being associated specific physical parameter). To ensure the overall model remains lightweight, we implement this design using a compact hypernetwork, enabling parameter- ization while retaining a model size comparable to standard, non non- con-paammtteriee FNOs.
Further, we extend the (parameterized) FNOs to effectively model the coupled systems while being efficient in the model size. While existing efforts (Xiao et al. 2023) introduce specialized decom- positions stream to capture cross-field interactions, there is little systematic guidance of the design principles that govern effective cross- varsall coupling in neural operators, particularly when aim- ing to preserve the efficiency and scalability of standard FNOs. In this paper address that gap: we present a principled design space for extending FNOs to coupled problems, investigate the lift, Fourier layers, and projection operators for identifying effective locations and mechanisms for en- abling cross-vares cross-variable variable coupling.
In addition, introduce
benchmark PDE based on simplified one-dimensional capaci- tively coupled plasma (CCP) model. This system is of practical importance in plasma physics and semiconductor manufacturing, and it exhibits rich parameterized dynamics that challenge existing operator-leaming methods. Our formulation provides a tractable yet representative setting for eval- uating neural operators under coupled and parameterized conditions.
Our contributions are summarized as
Parametric extensions of FNOs for effectively modeling parameterized dynamics, Systematic dissection and adaptation of FNO architecture components for coupled systems, Introduction of novel benchmark problem with many physical parameters describing sim- plified sheath dynamics governed by plasma physics, and
Extensive experimentation with the proposed methods, comparisons with baselines, and validation on benchmark problems.
TECHNICAL BACKGROUND
2.1 NEURAL OPERATORS
NOs are type of data-driven surrogate model designed to learn mappings between functions rather than traditional input-output pairs, making them particularly effective for problems governed by PDEs (Lu et al.) 2021 Kovachki et al.) [023] Boullé & Townsend, 2024, Azizzadeneshelli et al.) 2024). By taking discrete representations of continuous functions as input, NOs produce functional representations that can efficiently approximate and simulate complex physical systems.
Let A and be two function spaces, and consider nonlinear operator G A + U mapping between them. Neural operators are class of machine learning methods designed to construct surrogate Ge ss G within a trial space of neural networks. Given input-output pairs {(a', w²) w²) where z' A and u² G(a') U1 a neural operator can be trained in the standard supervised manner by solving: mine E L (Go(a'), u²), where L denotes loss term and θ denotes the parameters of the neural operator Ge.
In practice, the functional input data at are first discretized; for example, if a(x: and u(z) are functions, the grid representation a = [a(x1),
(,),),,............... a(xm)) )1" E jmmx serves as input along with an evaluation point (or set of points) D [xI,.... 3m) in the domain of a, and the neural operator leams a mapping a + (a that predicts (a) so u(xx) (Lu et al. 2021 et al. 2021). With the assumption that the spatial grid where a(x) and u(z) are discretized is fixed, (D.a) + (under G) produces a vector u of operator evaluations, and therefore samples (z, a, u) form the training data for the NO learning problem.
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2.2 FOURIER NEURAL OPERATORS
FNOs are a specific instantiation of NOs that model learnable nonlinear kernel integral operators, with the integrals efficiently evaluated in the Fourier frequency domain. FNOs, Ge : U, first projects the input data into hidden representations via local transformation, vo(2) P(a())), where P is typically chosen as a linear projector in practice. FNOs then employ the iterative updates of hidden representations Ug
c2+1 via
Ue+1 (I) = ( (K(a
Vo D
where W is a linear transformation and IC denotes a kernel integral operator. The kernel integral operator is defined as
(K(()))))))( (x) = S. k(x,, a(x), a(v): o)
Vx E D
(2)
where & indicates the learnable parameters that characterize the kernel. For efficient computation of Eq. equation?? 2] FNOs define the kernel integral operator Fourier space and perform a convolution operation in that space such that
(z) (Fee) (z).
Vo D,
where F and F-1 denote Fourier and inverse Fourier transformations, respectively, and Rg denotes Fourier transform of function k. In practice, Rs is parameterized as learnable tensor. The last hidden representation (x) is then projected back to the target data space via local transformation, u(x)= = (x)),, where Q typically modeled as shallow multi-layer perceptron (MIP)..
3 RELATED WORK
Building the success of FNOs, there have been many variants extending their capabilities. For example, physics-informed neural operator (PINO) (Li et al. 2024) and Geo-FNO (Li et al. 2023) extend FNO with physical consistency geometric generality.
More closely related to our work are efforts that enhance FNO architectures to improve expressivity and efficiency. Factorized FNO (Tran et al. 2023), U-FNO (Wen et al. 2022), and U-NO (Rahman et al. 2023) propose structural modifications: separable spectral layers or -ncoddrr designs that enable stable training. Our approach follows similar philosophy of architectural enhancement, but with emphasis on parameterized and coupled systems rather than general-purpose scalability.
Within the FNO family, explicit modeling of physical parameters remains relatively underexplored. The recently proposed HyperFNO (Alesiani et al.) 2022) infers the model parameters of FNO via hypernetwork conditioned on known physical parameters, illustrating the benefit of encoding para- metric dependencies. Another complementary study (Subramanian et al. (023)) explores scaling laws and transfer behavior of neural operators under large datasets and compute budgets. Our
method shares the goal of parameter-aware modeling but focuses on a compact modulation mecha- nism that preserves the lightweight character of standard FNOs.
For coupled systems, prior works such as coupled multiwavelet neural operator (CMWNO) (Xiao al.) (202)) and attention-based multiphysics models (Rahman et al. | 2024) have introduced cross- field coupling through wavelet large-scale pre-training. In contrast, our work provides sys- tematic and minimal architectural extension to FNOs, achieving efficient cross-variable interaction without sacrificing model compactness.
METHODS
Our methods seek minimal yet principled architectural modifications to FNOs that enable them to handle parameterized and coupled dynamical systems. We emphasize minimal modification design principle: this ensures (i) the resulting models preserve the efficiency and scalability of FNOs and (ii) performance gains be attributed to structural insights rather than increased capacity.
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To this end, we focus the three computational components of FNOs: the lift operator (input projection), the Fourier layers (spectral updates), and the projection operator (output mapping). We systematically explore how to adapt each component to encode parameter dependence and to capture cross-variable interactions in coupled systems. Section [4. presents the parameterized extension via (npernetwrrkbbssed modulation, and Section [.2 details the coupled extension by introducing the design spaces and spectral-domate coupling.
4.1 EXTENSIONS TO PARAMETRIZED FNO
compact hypernetwork- bbssdd modulation scheme is proposed to extend FNOs for handling
physical parameters, jr TR A lightweight hypernetwork (Ha et al. (017) takes the physi- cal parameters μ as input and outputs a set of shifts, s(z, μu) = fasper (z, μ)), where s(x, ()) =
[s (z, , St (z, u)] corresponds to layer-specific adjustments. These shifts are incorporated as additional biases in the Fourier layers:
((+ (2) a W (e((x)= (K(a; ()vz) (x) se(2, Vz D
(4)
We refer to this variant using "hp" (e.g., hDFNO) indicating the hypernetwrrk- bbsed approach (Fig- ure I.
The rationale behind this design is twofold. First, modulation has already proven effective in im-
plicit neural representations (INRs) (Sitzmann et al. 2020 Fathony al. 2020 Dupont et al. 2022) and physics-informed neural networks (PINNs) (Cho et al. (023). where conditioning on auxiliary variables enables networks represent families of functions without retraining. Second, unlike simple input augmentation, modulation operates directly on internal representations, allowing pa-
rameter dependence influence the dynamics at every layer rather than only at the input. Together, these properties make harvaarant lightweight yet expressive model for parameterized dynamics.
Compared to HyperFNO (Alesiani et al. 2022). which infers a large part of the base FNO through a hypernetwork, our approach adopts lightweight shift modulation that perturbs intermediate activa- tions while keeping the core weights fixed. Within the considered scenarios in this work, this design achieves comparable flexibility with substantially lower parameter overhead and complexity.
we also consider simpler variant that employs input augmentation to encode physical parameters. In this approach, the parameters are concatenated with the function input as [a(x); [(>)), where uux)) = І is constant across the spatial domain. The lift operator then encodes this augmented input into latent representation, (z) P([a(z), m(=)])) This strategy mirrors prior work, for example in the neural ODE community, where augmenting the input has been shown to enhance the expressivity of the learned dynamics (Dupont et al. 10199Massaooll et al. 2020 Lee & Parish| 2021), and PINNs (Cai et al. 2021 Cho et al. (2024). We refer to this variant using the prefix "p" (e.g.. pFNO) to denote parameterization via input augmentation.
For completeness,
While input augmentation is straightforward and effective some cases, it may not fully capture how physical parameters influence the underlying dynamics. The hypernetwork-basse modulation therefore provides a more expressive yet efficient alternative, conditioning the operator through- out its layers rather than only at input. Within our experiments, the hypernetwork-baat approach consistently achieves better performance without significantly increasing model complexity.
4.2 EXTENSIONS TO COUPLED FNO
We now turn to extending FNOs for coupled systems, where multiple interdependent variables evolve under shared dynamics. For simplicity of exposition, we focus on the case of two variables and denote each component by superscript, z for a generic variable 2, where O E {a, f}. With this notation, the operator of interest becomes (x)) uz(x) = Ge (a) a (x)). Our design philosophy follows the same principle outlined at the beginning of this section: to retain the mini- mal and scalable structure of standard FNOs while introducing only the modifications necessary to capture cross-variable interactions. To this end, we introduce coupling in the spectral domain, where the spectral representations of multiple variables are mixed through a lightweight encoder-decoder mechanism. To support this coupling, explore how the core FNO computational components, namely the lift operator, P. Fourier layers, and projection operator, Q, can configured, including choices whether those operators shared or defined separately across variables.
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Fourier layer
F
R
F=1
0(2)
W
F4,
Hyperretwork
5(4, x)
Figure 1: The proposed hPF architecture. The hypernetwork produces parameter--epeendntt 'shift' term, s(p, r), which acts as a parameter-dependent bias term in each Fourier layer.
Lift operator, P For the local transform, (P)) vo(2) Rd, consider main design choices that differ in whether the operator is shared or individual across the two variables.
(Shared): A single operator P R'' Rd is shared across the two input components, that both variables are mapped into the latent space using the same transformation. @ (Separate): Two individual mappings, P (Pa pO), allow variable-specific latent representa- tion.
@
Fourier layers (spectral coupling) The main modification enabling cross-variable interaction oc- curs in the Fourier layers, which consist of two components: the point-wise linear map (ey(z) (0 / (Q) and the global spectral convolution Kuxx (z) (G).
@ @ The point-wise linear map can be implemented either as a single operator shared by both variables (②) as two separate operators (W and W (())
④ For coupled systems, we define the global spectral convolution to perform coupling only in Fourier space. Specifically, each variable is first transformed independently. (P = Frr (k)), for O {a, Their spectral representations coder network fone (k) شه (k:), followed by the usual mode-wise kernel multiplication ö(k) Res (k)) The result is decomposed into variable-specific coefficients using another shallow net- work, (6 (k)), (k)) = fdoe (ok), and finally mapped back to the data space via the inverse Fourier transform, o = (k))(()))),, for D = o,
then combined through shallow
key design choice here is that coupling is introduced only in Fourier space, after the individual Fourier transforms. This has several benefits compared to approaches that use entirely separate con- volution operators and exchange hidden representations across variables. First, preserves the core structure of FNOs, keeping the model lightweight and scalable. Second, the Fourier domain nat- urally captures long-range correlations, making it a well-suited location for cross-variable mixing. Finally, by avoiding repeated exchanges of hidden states in the spatial domain, this design reduces computational overhead while maintaining symmetry between the two variables.
Projection operator, Q The local transform, (Q) u(x). resembles with that of the lift operator and, thus, similar choices apply. We focus here on the single-channel output format for each variable, which is the natural setting for coupled systems.
(Shared): A single projection is shared by both outputs, Q Rd. + Rd.
(Separate): Two operators are separately defined for each output component, Q = (Q" )*),
with Q Rd= + I for C E 3.
To refine the design space under @ we adopt the adaptive basis viewpoint (Cyr et al. 2020
which interprets the last hidden activation (a(x))) ('n))....,,,(... (a(x))] and the weight of the output layer as coefficients, Eat = .E...... En, T:
set of adaptive basis functions, V V( (z))
u(x) = Ei (x)) In practice, Q is parameterzed as a shallow MLP, (Qv7)(z) =
(oo(Wirrd bi) + by and the adaptive basis corresponds (((()) <Wiir(()) + b1)
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and the coefficients to E W2. The shared/separate choices then realized by sharing splitting the weight matrices and biases.
From this perspective, Q can differ in whether the basis and the coefficients are shared sepa- rate: @ Shared basis and separate coefficients, 12t Separate basis and shared coefficients, and Separate basis and separate coefficients.
4.3 PUTTING ALL TOGETHER
Based experimentation (presented in Appendix D.I.1 with the different combinations of the design choices (further elaborated in Appendix A). we define our model, extended FNOs - FNOx, realization of the combination: @
C 2 which employs the shared lift operator, the separate point-wing wise linear maps, the proposed coupled global spectral convolution layer, and the separate sets of basis functions and separate sets of coefficients the projection operator.
EXPERIMENTAL RESULTS
In this section, we present experimental results comparing the proposed methods with range of baseline models. All methods are implemented in PYTHON with PYTORCH (Paszke et al., 2019), building upon the original implementation of FNOs (Li et al.) 202T) and extending it handle coupled and parameterized settings. Each experiment is repeated five times with different random seeds, and all computations are performed on an NVIDIA A A.00 GPU with 80GB memory.
5.1 SETUP
Tasks and training We consider time-dependent PDEs that generate solution trajectories {u(æ, t; (o) with u(x; 0; μ4) u(0). NOs are formulated as one-step evolution maps that ap- proximate the discrete-time flow of the underlying PDE. Specifically, given finite history of past solution states (w(l- ([(--)))== ==o A, the operator predicts the next state u(l-1) E U: Ge A U. Multi-step forecasts are obtained autoregressively by recursively applying the same operator. Dur- ing training and testing, the initial input window (u(Tin - 1), ..... u()) is assumed to be available, while subsequent rollouts incorporate predicted states back into the input window.
Baselines
As baselines of comparisons, we consider methods inherited from Xiao et al. (202):
FNO,: FNOs taking inputs vertical concatenation of the discretized input data,
CFNO: Two separate FNOs sharing exchanging hidden representation Fourier layers,
MWT;: MWTs (Gupta et al. 2021 taking vertically concatenated input data,
CMWNO: Multiwavelet NOs specifically designed for coupled systems (Xiao al, 2023).
a FNO baseline considering parametric extension via hypernetworks (Alesiani et al.
2022):
HypezFNO.: FNOs whose model parameters of the lift/projection operators (P/Q), the pointwise linear map. W, and the kernel weight R are inferred from hypernetworks,
other relevant baselines, DeepONets (Lu et al. 2021 and U-Nets (Ronneberger et al. 2015), modi- fied handle multi input:
DON,: DeepONets extended to take vertical concatenation of the discretized input data,
U U-Net,: U-Nets taking two-channel input,
and, finally, one variant of certain combinations of choices from the described design choices:
FNO.: the lift operator projects the two-channel input (a a²) jointly into a shared latent space, i.e., P: R d with to (x: P([a (x), a (z)])-
We refer to Appendix B.1 for the detailed description of the baselines.
Performance evaluation We evaluate performance using the normalized root mean square error (nRMSE), averaged over all test trajectories:
nRMSE
4
0.7
2 0oT
(5)
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where and denote the ground-truth and predicted trajectories. For coupled systems, nRMSE is computed separately for each variable, and the final error is the sum: em nRMSE" + nRMSE*. For all methods, reported errors are computed on normalized fields; given the scale-invariant nature of the metric, the results are expected to be similar under the original physical sealing.
5.2 1-DIMENSIONAL CAPACITIVELY COUPLED PLASMA FLUID MODEL
In this section, we look at 1-dimensional capacitively coupled plasma (CCP) model as an example to demonstrate the performance of the proposed model in a parameteric setting.
The simplified CCP model describes www-empeertture plasma that is generated between two elec- trodes when an alternating voltage applied, operating in a manner similar to a capacitor, system, the oscillating electric field accelerates electrons, which then collide with neutral gas atoms or molecules, producing ions and additional electrons that sustain the plasma. CCPs are widely used in microclectronics manufacturing, including etching and thin-film deposition for semiconductor de- vices, as well as in materials processing and surface enginecring (Lieberman & Lichtenberg 1994 Rauf et al. 2023).
this
The governing equation is defined as follows:
m--6-0
0,6- V(t)
0, = - + R, O2e0 -e/co(n n2oo), (Poisson equation)
(Electron continuity equation)
Plasma
==
z= L
where ne(x, t) and 6(x, t) denote electron density and electric po- tential, which are the solutions of the equations, and & refers to a
Figure 2: The spatial domain
partial derivative with respect to O. F. is electron flux, defined
and boundary conditions.
diffusion flux and drift flux such as Ie = D0 - pm,0=Φ with electron diffusion coefficient D and electron mobility coefficient μu, and ion density is defined as fijo = Ro - z1) /mm/ee/ eTy R and Ro denote reaction rate and coefficient, respectively. The ion density is assumed constant in this model.
Physical parameters There are several physical parameters char- acterizing domain geometry, boundary conditions (BCs), and gov- erning physical dynamics (See Table in Appendix). Among them, in this study, we consider fixed parameters for geometry (illustrated in Figure 2). but varying parameters that define the boundary con- ditions and the dynamics. In the BCs, the zero boundary condition
Eleetron dimecty (m (m--))
given to the electron density (at x = 0, L). The electric poten- tial satisfies Dirichlet condition (at = 0), and time-periodic Dirichlet condition at r = L, (t) = Vo sin (2rr)..). That is, the po- tential the right boundary oscillates sinusoidally with amplitude Vo (driving voltage) and frequency f and we vary Vo in the exper- iments. This models the externally applied radio-frequency forcing that sustains the discharge in CCP. Along with Vo. we consider two other physical parameters that governs the dynamics: reaction co- efficient, Ro, and ion mass, m,. Figure3depicct solution snapshots of n. (x, t) and 6(x, t). simulated over single cycle, configured with Vo 100, Ro = 2.7 x 1020, and m; 40 x 1.67 x 10 10-27, 27,
Electric peteotial (V)
7,875
An example solu- tion snapshot of electron den- sity and electric potential.
Figure 3:
5.2.1 DATA SETUP
Data generation We collect solution snapshots by numerically simulating the governing equa- tions using finite-difference solver. The spatial domain is discretized into 128 cells, and the time domain into 100,000 steps per cycle, with the eycle determined by the boundary driving frequency of 13.56 MHz. fixed initial condition used to focus on scenarios where capturing physical dynamics is critical. We then subsample every 1000th snapshot, yielding 100 temporal indices per solution. Physical parameters are sampled on an equidistant grid, which contains 100 elements: Vo [100, 300), Ro E [2.7 2,7 yields 100 trajectories, each corresponding to one parameter configuration, which are then randomly split into training and test sets with a 9:1 ratio.
2.7 x 1020], and m, [1.67 x 10 6.68 10 26). This setup
10-26].
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5.2.2 NUMERICAL RESULTS
1-dimensional parameter space We perform three separate experiments, each on dataset ob- tained by varying a single physical parameter: Ro. Vo. or m,. Table [ summarizes the results, reporting prediction errors on the test set for the proposed methods and the baseline models. The proposed FNOx architecture (FNOx, without the parametric extension) provides notable improve- ment over the baseline methods. The overall second best model in this set of experiments is CMWNO and, compared to its performance, FNO,, achieve the prediction accuracy improved up to 38% (in the varying reaction rate scenario). The parametric extensions, pFNOx and hpFNOx, further improve the performance, achieving up to 55% improvement baselines. Compared to hyperFNO, which is also informed explicitly by the physical parameters through hypernetworks, the proposed variants, particularly, hpFNO, demonstrates significant im- provements (more than 40% for all cases).
the best performing non-parameterized
Table 1:
Performance comparisons between the proposed methods and the baseline methods. All models are tested on three parameters (Ro. Vo and m.) individually. The performance is measured in the relative error (nRMSE); the reported numerical values refer to the mean (+ std. dev.).
Model
Reaction rate
Driving voltage
Ion mass
FNOm
0.0403 (= 0.0012)
0.0791 (# 0.0243)
0.0363 (# 0.0067)
FNO
0.0375 (= 0.0055)
0.0873 0.0200)
0.0299 (+ 0.0030)
CFNO
0.0315 (= 0.0093)
0.0428 (# 0.0064)
0.0333 (+ 0.0076)
HyperFNO,
0.0278 (= 0.0117)
0.0355 (# 0.0160)
0.0253 (+ 0.0088)
MWT: 0.0409 (= 0.0032)
0.0639 (# 0.0038)
0.0403 (+ 0.0040)
CWMNÓ
0.0312 (= 0.0023)
0.0526 (# 0.0521)
0.0241 (# 0.0036)
DON-
0.0844 (+: 0.0058)
0.2147 (+ 0.0414)
0.1035 (+ 0.0174)
U-NET,
0.1084 (= 0.0345)
0.0844 (# 0.0354)
0.1719 (+ 0.1022)
FNOx
0.0193 (= 0.0059)
0.0345 (+ 0.0108)
0.0212 (+ 0.0062)
pFNOx
0.0194 (= 0.0075)
0.0278 (# 0.0053)
0.0142 (+: 0.0021) 0.0128 (+ 0.0017)
hpFNO,
0.0154 (= 0.0029)
0.0192 (+ 0.0040)
Computational/memory aspects Since prediction accuracy alone does not provide a complete picture of the experimental results, Figure 4 presents additional information under the varying driv- ing voltage scenario, including model size (4a) and computational timing (4b), alongside the mod- els' prediction accuracy. The model size is measured in the number of trainable model parameters and the timing reports a per-epoch training time measured and averaged across the total number of epochs. The both figures provide the same observation: the proposed methods FNOx, pFNO,, and hpFNO, achieve improvement in accuracy without sacrificing the model size (i.e., memory footprint) and the computational wall time too much. Additionally, Figure 4c shows the test loss trajectories over the training epoch. Along with the computational timing per epoch (Figure(4b). the loss trajectories provide complete picture on models' computational requirements in achieving the presented accuracy.
Study on varying Tin Although the true dynamics depend on physical parameters, all non- parametric NOs in Table achieve reasonable accuracy. This is because the NOs take as input window of past snapshots, characterized by Tm. akin to delay embedding in dynamical systems (Tak- ees 2006). By consuming multiple consecutive time steps, the NOs implicitly encode information about system parameters through temporal correlations, enabling them to infer dynamics.
Thus, the benefits of using the parameterized version of FNO,,can be more pronounced when Ti becomes smaller. To see this, we extend the experimentation for varying Tin = {10, 5, 2, [ Ta- ble 2 reports the results comparing the performance of FNO, and FNO,, and their two parametric variants (pFNOc, hpFNCc), and (pFNO,, hpFNO,), respectively. As Tin decreases, the overall model performance, which is expected. For Tin {10, 5). the improvement achieved by hpFNO, over FNO, over 20%. For Tin (2, 1), while the hp variants maintain the accuracy level around
For Ti (2,1) training instability was observed across some random seeds; due to this, trained models with 10 random seeds and report the average over the five runs with the higher prediction accuracy.
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rNOo
CTNO
pINOX
MWT
DONc
perferanne (AAKS))
0.175
mode zerformene In0ACCE)
109
mocel performanc
0.325
0.300
0.075
0.000
0.025
0.025
0.58
3.5
170
100
200
300
400
500
number of paramerars (x10%)
Eming Isec)
Epoch
(a) Model size
(b) Timing
(c) Test loss
Figure 4: [ID CCP] Plots depict the number of model parameters versus the model performance (left), computational timing versus the model performance (middle), and the test loss trajectories over epochs (right). These results are from the varying driving voltage scenario.
13~116% relative errors. Considering that only initial conditions are available in most practical se- tups, maintaining this level of performance could be considered important step forward for FNOs.
Table Performance comparisons of non-parameterizee (FNOc, FNOx) and parameterized (pFNOo, hpFNOc, pFNO,, NOFNO,)) models for varying Tin {10, 5, 2, 1}. All models tested on the reaction rate case. The performance is measured in the relative ee-error; the reported numerical values refer to the mean (#: std. dev.). indicates that the models fail to learn the dynamics and indicates that the model performance are averaged from the best runs out of 10 total runs.
Model
Ti 10
Th
The 2
Th
FNO.
0.0375 (#0.0055)
0.1324 (=0.0304)
1.0048 (10.6356)
1.4484 (+0.3128)
pFNO.
0.0334 (=0.0101)
0.0923 (=0.0276)
0.2151 (1000034))
0.3757 (#0.0679)
hpFNO,
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)
1.2832f (+:0,4298)
1.8143t (+00.558)
pFNOx
0.0194 (=0.0075)
0.0464
(=000158)
0.1640
(=0.0155)
0.2522
1376)
hpFNOx
0.0154 (#0.0029)
0.0317 (=0.0022)
0.1324 (+:0.0242)
0.1372 (+000100)
5.3 GRAY-SCOTT EQUATIONS
As the second benchmark problem, we consider the Gray- Scott equations, which form a system of coupled reaction- diffusion equations describing the spatio-temporal dynamics of interacting chemical species. We consider parameterized dynamics of this equation, where the parameters are diffusion coefficients and feed rate. The task of the NOs are perform- ing predictions of time-dependent PDE solutions in the auto- regressive manner, and we follow the same training/testing protocols shown in the first benchmark problem. We refer to Appendix for for details (the equations, the experimental setups, and additional results).
Performance compar- isons: The performance is mea- sured in the relative f--error; the reported numerical values refer to the mean (# standard deviation)
Table 3:
Model
Performance
FNO
0.01 + 0.0026)
FNO=
0.0097 + 0.0030)
CFNO
0.0161 (+: 0.0045)
MWT.
0.0092 (+: 0.0021)
CWMNO
0.0293 ( 0.0078)
DON. 0.0138 (+ 0.0020)
Table summarizzes the performance of the proposed and base- line models the Gray-Scott benchmark, where the task is to predict the spatiotemporal evolution of two reacting chem- ical species under varying feed-rate conditions. The proposed
0 U-Nete
0,0159 (+ 0,0071)
FNOx
0.0075 ( 0.0016)
pFNOx
0.0041 ( 0.0010)
hpFNOx
0.0022 ( 0.0006)

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variants consistently outperform all baseline methods, achiev-
ing relative e -rrrroos of 0.0075, 0.0041, and 0.0022 for FNOx, pFNO,, and hpFNOxx respectively. Among the baselines, the multiwavelet-bssed model (MWT,) performs best with 0.0092 error, yet the parameterized versions our model reduce this by more than half. These results indicate that incorporating parameter conditioning and lightweight coupling in the Fourier domain substantially improves predictive accuracy while maintaining the compactness of the original FNO architecture. Overall, the apFNOx variant achieves up to 72% lower error compared to the strongest baseline, demonstrating effective generalization across parameterized reaction-diffuse diffusion dynamics.
6 CONCLUSION
This work has investigated extensions of Fourier neural operators (FNOs) for modeling parameter- ized and coupled time-dependent partial different equations. To enhance the capability of FNOs in handling parameterized dynamics, we have proposed novel modulation-based architectural modi- fication that incorporates parameter information systems, we have defined design spaces of FNOs and conducted an in-depth systematic investi- gation identify economic architectures that preserve efficiency while capturing inter-component interactions. In addition, we have introduced novel benchmark problem on plasma physics, which introduces rich parameterized dynamics and contributes a valuable testbed to the community. Exper- imental results have demonstrated that our proposed approaches achieve significant improvements in predictive performance, while maintaining model size and runtime efficiency comparable to stan- dard FNOs.
lightweight yet effective manner. For coupled
7 ETHICS STATEMENT
This study focuses methodological advances in machine learning for computational physics prob- lems, and we do not anticipate any significant ethical concerns arising from the proposed approaches.
REPRODUCIBILITY STATEMENT
To support reproducibility, we provide detailed descriptions of the proposed methods, including model architectures, algorithms, loss definitions, hyperparameters. training setups, and baseline comparisons. All experiments are described in the main text and appendix, and upon acceptance we will release the full source code to further facilitate reproduction of our results.
9 ACKNOWLEDGEMENT
The authors acknowledge funding support from Applied Materials Inc K. Lee acknowledges partial support from the U.S. National Science Foundation under grant IIS 2338909. K. Lee also acknowl- edges Research Computing Jennewein et al. (202)) at Arizona State University for providing HPC resources that have contributed the partial research results reported within this paper.
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A DESIGN SPACES OF FNOS
A. DESCRIPTION ON EACH COMPONENT
Data space
Separate single-channel format. Each variable is treated as an independent function, a(x) (x), a (a)),, and discretized separately as a a a E R". This representation keeps the two components distinct throughout the architecture.
i Two-channel format. The input is represented as vector-valued function, a : -> R2, - (x), a (z)), which in a discretized form becomes a = =Ia", a²] Rmxx The output is rep- resented analogously as u(x) = (x), u Rmx2
ux)) (z)]² E I2a and the discretized form u u u
Lift operator, P
? The lift operator P R2 Rd and shared across the two input components, so that both variables are mapped into the latent space using the same transformation.
Alternativel, the lift operator can consist of two separate mappings, P (Pa P*)) each specific
to one input component. This allows the two variables to have independent latent representations from the beginning.
@ Compatible with @9 the lift operator is defined as P IR22 -> Rdd, mapping two-channel input hidden state.
Fourier layers
! The point-wise linear map is single operator shared by the both variables.
L The point-wise linear map is defined as two separate operators (Wo, W.).
C For coupled systems, define the global spectral convolution to perform coupling only in Fourier space. Specifically, each variable is first transformed independently. aF (k). for 0 = {a, B}. Their spectral representations are then combined through a shallow en- coder network fen ())(() o (A:),, followed by the usual mode-wise kernel multiplication ö(k) = (k))). The result is decomposed into variable-specifie coefficients using another shallow net- work, (6*(k), (K) = fdee (ū(k). and finally mapped back the data space via the inverse Fourier transform, ( (b) (k)()),)) for a, 3.
c The standard global spectral convolution performs the Fourier transform, mode-wise kernel mul- tiplication, followed by the inverse Fourier transform: FF Rs (Fuz) (x).
Projection operator, Q
OI The projection operator shared across both output components, Q Rdd R
JZ The projection operator is defined separately for each output component, Q (Q", Q?), with Q dd + R for E a, 3. From the adaptive-basis view perspective, Q can differ in whether the basis and the coefficients are shared separate across the two variables:
Shared basis and separate coefficients, and
J2L Separate basis and shared coefficients, and
Separate basis and separate coefficients.
TG Similar to @. this projection operator is defined as Q Rdd. + RR?,, mapping a hidden state to two-channel output.
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A.2 SUMMARY OF THE COMPONENTS AND RESULTING MODELS
Table 4 summarizes the architectural components and configuration options discussed in Sections 4.1-4.2 and Appendix A.I, highlighting how shared separate design choices be applied to handle parameterized dynamics and multi-variable (coupled) systems.
Table Summary architectural components and configurations explored in FNOx.
Componeut
Design choice / notation
Purpose and interpretation
Parameterization
Input aug.entaation (pENC)
Introduces explicit dependencce on the physical paramcter In the hypemerwork-hasee appreach. small nerwark ocipets layer-wise shifts ec(i. p)) that modulate activations without regenerating full weights.
Syypereewwrr--heee modula- tion (bpPNO)
operator,
(Pt) Separate (F2)
space. multi multi-variable settings, operator uses the same transfommation for all variables, while separate one allows variable specific representations.
Projects input
across variables
map,
(L, / Separate (Lgg)
tramsfomaaions in tains parameter efficiency. while separating allows each variable to evelve differently
Controls local
layer. Shuring
acress variables
space.
Global spectral convolu- tion, G
Standard (G2) Compled
Defines noe-local interactions Fourier space. spectral representations of multpplc variables via au encoder-decoder, ca- abling variable interaction within single openaor.
The coupled variant mixes
Projection operator, Q
Shared (Q1) Separate hasis
Maps latent states back the physical separate buses and coefficients lets euch variable form its own representation while improving expressivity.
In coupled systems, using
and coefficients (Qaa,h,))
Now we list realizations of FNOs obtained from some combinations of the above design choices.
FNO,: ② @ i it
@
two-channel input is processed by standard FNO to
produce a two-channel output.
FNO,: @ + 1 +
@
@ single-channel variables concatenated in a discrete
representation, i.c., ja a T p22m the concatenated input is processed by a standard FNO to produce single-channel errically-comcaatenated output, i.e., [u" u FNOx: This is class of the proposed models, which can be realized with the combinations of
2m
2 In the experimentation, with FNOx, we refer to unless otherwise stated.
A.3 VISUAL COMPARISONS OF ARCHITECTURES
We provide of visual representations of the proposed architectures illustrate the modifications made in this work and highlight the difference from the original FNO architecture (Figure
<(2)
u(z)
PRO
a(x)
P
Fourier layer
Fourier layer
Fourier layer
u(x)
Fourier layer
F
F
x(x)
W
Figure 5: The original FNO architecture
Parametric extension In the first set of illustrations, we present the visual representations of the parameteric extension of FNOs: pFNO and hyFNO. Figure depiics the PFNO architecture, which
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augments the physical parameters with the input function, which is then used the input to the FNO layers.
a(x)
P
(u(2)
u(x)
(a) FNO architecture
a(rr)
a(
P
vD(2)
u(a)
Porancter
(b) PENC architecture
Figure 6: The abstract representations of (a) FNO and (b) pFNO architectures are depicted.
Figure illusttrates the additional computational flow from the physical parameter the shift func- tion 6(1, x), which then acts as the bias in the Fourier layer.
Fourier layer
F-1
v(z)
W
f1,
HyperNetwork
s(M, x²)
Figure 7: The proposed hpFNO architecture. The hypernetwork produces darameter-depenndnt 'shift' term, s),, which acts parameter-dependent bias term in each Fourier layer.
Coupling extension
The next of visual representations explains how the proposed architecture models coupled physi- cal behavior. Figure illusstrats the design spaces for the lift operator P and the projection operator Q. These two components determine how information is encoded and decoded across variables. In multi-variable systems, P controls whether the input fields share the same latent representation retain variable-specific mappings. while Q defines how the learned latent features projected back into each physical field. Figure 8b further illustrates three possible configurations of Q (i.e., shared basis and coefficients, shared basis with separate coefficients), and fully separate basis and coefficients. The light orange blocks indicate shared operators, whereas the darker orange blocks indicate variable-specific operators.
Figure 9 presents graphical overview of the proposed coupled extension of the FNO architecture. The coupling mechanism is introduced through an encoder-decoder pair that mixes the spectral representations of multiple variables directly in Fourier space. The encoder aggregates information from the transformed variables, and the decoder distributes the coupled representation back to each channel. enabling variaale interaction without substantially increasing model size. This design preserves the efficiency of the original FNO while introducing controlled coupling through a single shared operator.
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a(*)
Fourie layer
Fourier layer
Foune layer
u(xr)
(x)
P
(=t)
Q
("(r))
(2
4"
(x)
(m)
a
p8
(2))
Q
(x)r)
(E(
Q
( (=)
(a)
PNOx architecture
(x))
W2
2(x
WF'
(2(())
W2
(x)
W
(
W
(x)
W
W
(
(b) The design choices for Q
Figure The standard and the proposed FNO architecture for modeling coupled behavior.
Coupled layer
(a)
F-1
+
(2)
F-1
Figure 9: The proposed coupled extension of FNO architecture. The @ symbol indicates the con- catenation operation.
B IMPLEMENTATION DETAILS
DESCRIPTIONS AND SPECIFICATIONS OF THE BASELINES AND THE PROPOSED MODELS
FNO.. FNO. FNO, For all FNO-based baselines, we consider the same Fourier layers with the modes and width (k: 12 and d, 20). P is parameterized as linear layer and Q is parameterized as a shallow MIP with one hidden layer. All nonlinearity functions ReLU (Nair & Hinton 2010 This fixed hyperparameter setup is to control the representational capacity contributed by these hyperparameters. Consequently, any observed performance gains can be attributed to the architectural modifications introduced in each variant rather than to differences in model capacity.
For defining the FNO,. consider the combination, 2 + + Jay which employs the shared lift operator, the separate point-wise linear maps, the proposed coupled global spectral convolution layer, and the separate sets of basis functions and separate sets of coefficients in the projection operator. The encoder/decoder, fdee, in the Fourier layers set to simple linear layers. This choice is obtained from the ablation study.
PFNO,/ pFNO, These parameterized variant takes the standard input (i.c., the input to FNO, and FNOx), but augmented with the physical parameters, J..
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hpFNO,/ /hpFNO, These variants require additional component, the hypernetorrk, fryrer In our implementation, we consider a simple linear layer to parameterize the hypernework, following the design used in (Dupont et al. 2022), and model the hypernetwork to take (a, μ)) along with the discrete reprsentation of the windowed history, { {u(t - ? Thus, the hypernetwork becomes local transform similar to the lift operator.
HyperFNO. HyperFNO (Alesiani al., 2022) integrates a base FNO with a hyper-network; the hyper-network takes the PDE parameter vector as input and generates FNO's weights. For each Fourier layer É, the hypernetwork generates a subset of the FNO parameters such as the pointwise linear map (Wz), the spectral convolution weights (Rş), and the lift/projection operators (P and Q), according to
We hw (w: @w), Re [R((;(R), P= hp(w:pp)
Q ha (1: 0Q).
(7)
where hn 0-) for O (W, R, P) denotes hyper-network with its own model parameters 8..
To control model size, two variants proposed: the Addition form, which scales existing weights by p-ddpennent factors, and the Taylor form, which further factorizes the modulation to reduce parameter count.
In the Addition form, the hypernetwork modulates the base FNO weights through low-rank param- eterization:
hn(02 p1) R/ + (V& (VZdiag(hnna V2 Orow.ol Rí, hw (0 p1) W6 - (Uw (w,( Drow W
(8)
where Crow and Crow.ol denote row-wise and element-wise modulation, respectively, and (ي refers to shallow MLP. In our experimentation, we use a two-layer MLP with width 128.
In the Taylor form, the modulation is expressed through a first-order expansion around the base parameters:
R (0 p)) R/ (T+ V diag((h,,, ) hw 11) W6 @ (T+ Uhr (w))
where denotes element-wise multiplication.
In the experiments described in the main body, we consider the Taylor variant; in the Appendix, we provide the results of both variants. To differentiate those two methods, we denote them by HyperFNo,-T -T and HyperFNO.-A.
MWT./ CMWNO The multiwavelet- operator learning framework (MRT) has been intro- duced as approach to leverage multiwavelet expansions for differential equations. By decompos- ing solution operators into multiwavelet bases, MWT captures both global structures and localized variations in flexible and efficient way, capable of handling multi-scale dynamics and heteroge- neous patterns. The coupled extension of this has been introduced in the sequel (Xiao et al., 2023) which is named as MWT, and is designed to take vertically concatenated input variables.
Extending this idea, the coupled multiwavelet neural operator (CMWNO) adapts the MWT framework to systems of coupled partial differential equations. CMWNO introduces a strategy that decouples the operator kernels across different wavelet components during decomposition and reconstruction. This enables the model to efficiently learn and represent the interactions between multiple coupled fields while keeping computation tractable. Computationally, CMWNO consists of two distinct operators that communicate by exchanging hidden representations. A randomized scheduling mechanism, referred to as the 'dice' algorithm, determines the order in which these operators share information, enabling flexible and efficient coordination between the coupled components.
For both MWT and CMWNO, we use the implementation available from the Github repository The scaling parameter a is set to 12, the channel multiplier c is set to 16, the wavelet polynomial order is set 4, and the base polynomial basis is set to the 'Legendre' polynomials. The dice strategy
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is utilized follows: for each batch throw random number from uniform distribution [0,1] and if the sampled random number is below 0.5, the second operator takes the first operator's 's hidden representation through the roll-out. On the opposite case, the first operator takes the first operator's hidden representation through the roll-out.
CFNO CFNO mimics the operations of CMWNO, i.e., two separate operators enabling coupling effect via sharing a hidden representation of one operator to another. For CFNO, the backbone is FNO instead of CFNO. Also, the same dice approach described above is utilized.
Same as other FNO- baselines, we consider the same hyper--aramenneee parameter 4 Fourier layers with the modes and width (k 12 and dy 20). P is parameterized as shallow MLP with one hidden layer. All nonlinearity functions are ReLU, For CFNOs, we have two separate realizations of FNOs and in the third layer of CFNOs, the hidden representation from one CFNO is passed to another CFNO, where the direction is dictated by the dice approach.
linear layer and Q is parameterized
DeepONets For the implementation DeepONets (DONs), we base our implementation on a publicly available PyTorch version. We modify the DONs to take multivariate input and produce multivariate output. We explore the combination of a (shared/separate) trunk net and (shared/separate) branch net, and find that two separate branch networks and a shared trunk network perform the best. Each branch network takes the same input, which is concatenated time windowedd snapshots; then one branch network produces coefficients for the electron density and another branch network produces coefficients for the electric potential. That is,
n ne(x,,+11)
branch, (ā-se) ( Stun (I),
(10)
Φlz, 1 (+ branch, (āc:),
( (x)
where gbranch.. and foranch,, denote two separate MLPs, taking time-windower input
(2, ((1,))
ne(2, / Tin +1) o(x, - Tı + 1]
(11)
where Tin decides the window length. The trunk net is modeled as separate network, producing two separate sets basis such that
prrunk, (x), f ftrunk (x:) = [W ptrunk (x) + 6" y (x) + b"d,,
(12)
where finnk denote shared MLP, and (W )"r) and (W bn)) denote a set of model parameters specific to the electron density and electric potential, respectively.
For each branch networks, we use 4 hidden layers with 1024 neurons in each layer. For the trunk network, we use 3 hidden layers with 1024 neurons. For both network types, we consider ReLU for nonlinear activation.
U-Net For the implementation of U-Nets, we base our implementation on the code base we use for DeepONets. We modify the U-Net to take 1 1-dimensional and 2 channel inputs, and produce the output with the same dimensional specification. The U-Net architecture we consider has an encoder-decoder structure with skip connections.
The encoder progressively reduces the spatial resolution while increasing the feature dimension. begins with initial block (DoubleConv ReLU), followed by four downsampling stages (Max- Poolld - DoubleCony - ReLU), where each stage halves the sequence length and expands the channel dimension. The spatial resolution is halved at every downsampling and the channel dimen- 128)..
sion is updated
The decoder mirrors this process with four upsampling stages (Upsample - Concat (skip- connection) + DoubleCony - ReLU). At each stage, the feature map is upsampled by a factor of two, concatenated with the corresponding encoder feature (skip connection), and refined by double convolution block. Finally, the output layer applies ID convolution to project the features to the desired number of output channels.
TThis accessible through http:////mmm..eoooomooccc/c///am
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B.2 TRAINING DETAILS
All FNO variants and wavelet-based operators are trained for 500 epochs using the Adam opti- mizer (Kingma & Ba 2013) with a learning of 0.0025, weight decay 0.0001, and mini-batches of size TO. For U-Nets, we consider the same hyperparameters. except for the total training epochs, which is set to 1,000, which is to mitigate the effect of the increased model sizes. For DeepONets, we again consider the same hyperparameters, except for the learning rate, which is set to 0.0001 and the total training epochs, which is set to 2,500 which to compensate the decreased learning rate. For all baseline models, we repeat 5 different runs for varying random seeds. For all methods, we apply a linearly decaying temporal weighting to the rollout loss, emphasizing short-horizon pre- diction accuracy while progressively reducing the contribution of longer-horizon errors to mitigate compounding autoregressive effects.
B.3 ORTIIOGONALITY LOSS
To measure the orthogonality of the learned basis functions, V((x:) [x))......... Wn, (2)], we define a discrete inner product over the spatial domain. For a spatial domain and a measure p(x), the continuous inner product is (vi, wby) = fD va (x)dp((()) In practice, we approximate this inte- gral using numerical quadrature. Specifically, we employ Gauss-type quadrature rules, which select quadrature nodes Ty E D and associated weights wK. yielding (w/s, 15)) = Σm= Wx (1x))))))))))))))))))) (mx)-
With this discrete formulation, we compute the Gram matrix, [G] = (uh, bby), , for i.j = 1...., ml and quantify deviation from orthonormality via Lutho |G In, ll= leading the ultimate loss = Luse + Aontho This approach allows for principled enforcement of orthonormality in the learned basis while ensuring consistency with the underlying continuous inner product and measure. For Aurtho we consider 10.1, 0.01 0.000..
C ID CCP: DESCRIPTION
C.I GOVERNING EQUATIONS
The simplified one-dimensional capacitively coupled plasma physics is described by two equations: electron continuity equation and Poisson equation. The electron continuity equation describes the conservation of particles; electron density changes in time because of electron flows out the sur- face and because they can be created/destroyed locally due to the chemical reactions. The Poisson equation relates how the electrostatic potential varies in space to the distribution of charges (i.e., electrons and ions) in the plasma. The governing equations are defined as follows:
Ôn. or.
+ R
(Electron continuity equation)
Ot
(13)
e
Or = (ne - n.))
(Poisson equation)
60
where ne( t) and o(x, t) denote electron density and electric potential, which are the solutions of the equations, and oh refers to partial derivative with respect to D. Т. is electron flux, defined as diffusion flux and drift flux such as
Te = - Dāgn, - pm.020
(14)
with electron diffusion coefficient D and electron mobility coefficient μ. which is defined as = eTe/meum. Here, μ e/m.0m.- Here, Te, me, and 9,, denote elementary charge, electron temperature, electron mass, and collision frequency. The ion density is defined as nio = Ru(22 z /m/eTTT. where R(x) and Ro denote reaction rate and coefficient, respectively. The reaction rate is a spatially dependent quantity such that
{ Ro- 0,
x E [21, - x2, L - x1] otherwise
R(x)
(15)
The boundary conditions are specified as
The 0, 0,
at r 0,
(16)
Ne = 0, = (t)
at r = L,
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where V(t) Vo sin(2 That is, the zero boundary condition is given the electron density
(at x = O. The electric potential satisfies a homogeneous Dirichlet condition (at x = 0), and time-periodic Dirichlet condition = L. That is, the potential the right boundary oscillates sinusoidally with amplitude Vo and frequency f-
Table 5 lists up the important parameters, their description, and their values. Other important
stants are elementary charge, e = 1.6 x 10-19 19 (C) and vacuum permittivity co 8.854 x 10 10-12 (C*kg "c-kg-1m-3 -m-3)))
Table 5: Input parameter symbols, descriptions, and values
Symbols
Descriptions
Values (unit)
Domain Length
0.025 (m)
21, 22
Reaction area
0.005(m), 0.01
f
Driving frequency
13.56 (MHz)
Vo
Driving voltage amplitude
[100, 300] (V)
Ro
Reaction rage coefficients
92.
2.7 x x10 (m 35-1)
To
Electron temperature
(cV)
Ion mass
[1.67x10-2] 26 6.68 100
[0002]] (kg)
me
Electron mass
9.109x 10-30--0 31 (kg)
In
Collision frequency
10a (s-
C.2 VISUALIZATION THE SOLUTION SNAPSHOTS
To provide better understanding on the dynamies of the benchmark problem, we present some col- lections of solution snapshots for varying input parameters, reaction rate (Figure (T0) and driving voltage (Figure II] in the first RF cycle. All Figures presents the two spatio-temporal fields, the electron density (2, f) and the electric potential 6(x, t), on a plane, in a heatmap format.
Figure 10 shows the solution snapshots for varying reaction rates Rg = (2.7 x 10th, 1.485 x 20 2.7 x 10a0] 3s- The reaction rate serves quantity that determines how fast ion- ization occurs. The reaction rate drives the plasma density and sustainment in a CCP discharge. The higher the reaction rates, the faster the ionization is, and the higher plasma density is. Reaction rate can be controlled using power in plasma processing systems. Figure T0 shows some example solution snapshots for varying reaction rate; from the left panel to the right panel, the reaction rate increases, which results in higher electron density and thinner sheaths.
Elcetron deraity (mo ?)
Electron deneity (m) ")
Kleetron density (m.
0-
Electric poteutial (V)
Electric patmntil (V)
Electric poseusial (V)
0-
0
0-
7.874
2.50
(a) Ro 2.7 19
(b) Ro 1.485 x 10
(c) Ro 2.7 x 20
Figure 10: (ID CCP] Solution snapshots presented in heatmap: the solutions are collected for vary- ing reaction rates Ru {2.7 19 1.485 x 1020 2.7 10203.
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Figure 11 shows the solution snapshots for varying driving voltages Vo {100, 200, 300) in the first RF cycle. The driving voltage, given as the boundary condition, establishes the time-varying electric potential fields. The higher the driving voltage, the larger is the sheath voltage, resulting in stronger electric fields. The sheath is thicker at higher electric field. Figure 11 shows example solution snapshots for varying driving voltage: (a) Vo 100, (b) Vo= 200, and (c) Vo 300.
Electron detaitt (m
Electron deasity (m9 )
Electron density (m
0
Electric potential (V)
2002
0.025
Electric potential (V)
Electelc poseusial (V)
0-
0-
10
6.37
7
7.57
(a) Vo
100
(b) Vo 200
(c) Vo 300
Figure 11: [ID CCP] Solution snapshots presented in heatmap; the solutions are collected for vary-
ing driving voltages Vo {100, 200, 3003
Likewise, varying the ion-mass also provides variations in the dynamics. However, compared to other two physical parameters, the reaction rate and the driving voltage, changing the ion-mass parameter has slightly less physical impact on the dynamics and thus, the solution snapshots.
D ID CCP: ADDITIONAL EXPERIMENTAL DETAILS AND RESULTS
D.1 ID CCP RESULTS
In the following sections, we present additional experimental results that are not presented in the main body due to the space limit. The additional experiments include an ablation study on different design choices of FNOs, study effectiveness of the proposed hypernetwork and modulation based architecture for the parametric extension, and a study on extrapolation in the parameter space.
D.1.1 ADDITIONAL RESULTS
Continuing from the experimental results presented the main text (i.e., the "1-dimensional pa- rameter space" experimentation), we present the further detailed information regarding the com- putations: the model sizes and the computational timings, depicted in Figure [2] This time, the information obtained from the experimentation with the datasets that vary the reaction rate and the ion masses. Again, the model size is measured in the number of trainable model parameters and the timing reports a per-epoch training time measured
averaged across the total number epochs. We can make the similar observations here: the proposed methods FNOx, pFNCx, and hpFNO, achieve improvement in aceuracy without sacrificing the model size and the computational wall time. In case of the ion-mass scenario, two scenarios (the reaction rates and the driving voltage). This is largely due to the fact that vary- ing the ion- ion-mass provide less changing physical dynamics and taking the windowed history input, {u(t 7)] Th-1 allows other non-parameterized NOs to infer those physical parameters to some extent. Still, the proposed methods FNO., pFNO,, and hpFNO, improves the performance in that scenario while minimally affecting the model size and the training time.
relatively small improvements compared to other
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FlOm
CPNO
pfNOx
MWK
DONG
0.10
0.10
Rode performance (20A5))
performance (yRorse)
modbl aeformanca (1000SE)
0.08
0.08
0.04
0.02
3.5
100
200
500
400
500
number of model pa ramteess ox10")
tminng (oet)
Epoch
(a) Model size
(b) Timing
(C) Test loss
x
30"
moze perormance (68365))
0.14
perfomannee (mRISS))
mosal (MIMSE)
0.12
0.10
0.10
0.08
0.08
10
0.08
0.08
0.04
6.02
0.58
142
200
300
Eimnng (Scc)
Epoch
(d) Model size
(c) Timing
(I) Test loss
Figure 12: [ID CCP] Plots depict the number of model parameters versus the model performance (left), computational timing versus the model performance (right), for varying reaction rates Ro (top) and ion masses m; (bottom)
D.1.2 COUPLED EXTENSION
Ablation on model architectures To investigate the effectiveness of each design component, perform an ablation study on several different combinations of the design choices. We begin with the default combination, T ! + @. which consists of the shared lift/projection operators and the wisnt-wise linear map in the Fourier layers. We first consider replacing the point-wise linear map in the Fourier layer to be set of separate point-wise linear maps, Ta and W After that, we investigate the projection operator by testing three different combinations: Shared/separate basis functions V and shared/separate coefficients E. Finally, we test the effect of a of separate lift operators. Table 6 reports the results of this ablation study, The normalization shown in the last
Table 6: Performance comparisons of different realizations of FNO,. The models are trained for 1- dimensional parameter space, (reaction rate). The performance is measured in the relative ee-rrr (nRMSE); the reported numerical values refer to the mean (+: std. dev).
Variant
Description
Relative errors
Default setup
0,0341 (#: 0.0044)
Separate W
0.0281 (#: 0.0048)
Separate W and coefficients E
0.0259 (#: 0.0032)
+
Separate W basis V Separate W and basis V and coefficients
0.0315 (+= 0.0028)
0.0275 (+ 0.0062)
1
W. norm
Separate W and coefficients with layer norm Separate W and basis V coefficients E with layer norm
0.0205 (+: 0.0043)
norm
0,0193 (#: 0.0059)
norm
Separate and basis and coefficients with layer norm
0.0204 0.0035)
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Electron density (m-*)
X
Electron density (m--)
x10
0,025
0,025
FM0c
CPNC
7.87l-8
7.374-88
(a) pRMSE over 2/
(b) Pm 2 x 10
(c) 4 10
Figure 13: [ID CCP] (a) The model performance measured in nRMSE across varying Vm
2. 4) 108 (b) and (c): Solution snapshots collected for varying Vm {2 while keeping all other parameter values.
10 4 x 10
entries of the table is layer norm applied to the pre-activation values in the projection operator (i.e., W1 (x)r(x) b1). From this ablation study, we choose the combination, @ @+ F for all experimentation.
(2) with layer norm,
Varying collisional coupling To examine the effect of coupling strength in the ID CCP system, vary the electron-neutra collision frequency parameter V/n across 1 10 2 x 10 and 4 x 10a. As summarized in Figure [3a we observe general trend of increasing nRMSE with higher Vm across all models. While the proposed hpFNO, exhibits mild degradation the largest Vm, it still achieves the lowest overall errors (nRMSE x 0.015-0.018).
This behavior consistent with the underlying plasma physics: as the system becomes more col- lisional, electron transport is increasingly suppressed. In particular, increasing Vn strengthens the coupling between charged species and the background gas through enhanced momentum exchange, while simultaneously reducing both the diffusion coefficient D = eTe// (m.ym) ) and the mobility μ e/ (m. The reduced transport leads to stiffer solution snapshots with sharper spatial gradi- ents and weaker temporal smoothness (Figures 13b T3c. Consequently, the resulting fields exhibit steeper localized structures that are more challenging for spectral operator networks approximate.
eTe/(ma.m,
Taken together, these results suggest that higher-collisional regimes introduce both stronger coupling and increased stiffness, leading to moderately higher errors. We note, however, that it is difficult to fully isolate the influence of coupling strength alone, since changes in vm simultaneously affect both the degree coupling and the stiffness of the governing dynamics, thereby altering the learning difficulty and the numerical conditioning of the operator.
D.1.3 PARAMETRIC EXTENSION
Study on the effectiveness of the proposed model architecture To further investigate the ef- fectiveness of the proposed parameteric extension, compare the performances of the non- parameterized versions, FNO, and FNOx, against their own two parameterized versions, (pFNO,, hpFNO.) and (pFNCx, hpFNo,).
Table 7 reports the models' performance measured in all three physical parameter scenarios. Tak- ing the non-parameterized version the baseline, T the achieved prediction accuracy of each model and their improvements over the non-parameterized versions, presented in the per- centage. The results indicate that simply augmenting physical parameters input provide some improvements in prediction accuracy. However, the new architectural modification (i.e., hyper- network and modulation) brings significantly improved results (upto 68.01 % improvement). As reference point, we also report the performance of two variants of HyperFNO, and -A (Taylor and Addition) as well. The overall performance HyperFNc are in in-between those of pFNC, and hpFNO,. suggesting that the parameter-effiieent modulation technique (i.e., latent shift modulation) provides benefits in the given experimental contexts.
Study on the -iit---distribution samples In the main experiments, we follow the standard FNO testing strategy. In parameterized scenarios, however, explicitly sample parameters that
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Table Performance comparisons of non-parameterized (FNOc, FNO,) and parameterized (pFNO... hpFNOc. pFNOx: hpFNOx) models. All models are tested on three scalar parameters (reaction rate, driving voltage, and ion mass) individually. The performance is measured in the relative l² emror; the reported numerical values refer to the mean (# standard deviation, improvement over non-parameterized version).
Model
Reaction rate
Driving voltage
Ion mass
FNO,
0.0375 (+ 0.0055)
0.0873 (+ 0.0200)
0.0299 (+ 0.0030)
pFNO.
0.0334 (#: 0.0101. 10,94 %)
0.0454 (#= 0.0094, 47.92 %)
0.0284 (#: 0.0023, 5.08 %)
hpFNO.
0.0196 (# 0.0021, 47.57 %)
0.0279 (#: 0.0016, 68.01 %)
0.0210 (+ 0.0036, 29.84 %)
FNO,
0.0193 0.0059)
0.0345 (#: 0.0108)
0.0212 (+: 0.0062)
pPNOx
0,0194 (#: 0.0075, -0.51 %)
0.0278 (#: 0.0053, 19.30 %)
0.0142 (#: 0.0021. 32.91 %)
hpFNO,
0.0154 (+ 0.0029, 20.12 %)
0.0192 (+ 0.0040, 44.40 %)
0.0128 (+ 0.0017, 39.83 %)
HyperFNO,-1
0.0278 (+: 0.0117)
0.0355 (#: 0.0160)
0,0253 (#: 0.0088)
HyperFNO,-
0.0276 (#: 0.0119)
0.0401 (#: 0.0143)
0.0250 (#: 0.0098)
lie entirely outside the training range, which we define as out-of-distribution (OOD) testing. As benchmark case, we vary the reaction rate Rg. For training. 101 equidistance values of the reaction rate is sampled from 1 2.7 xx102';; for OOD testing, the interval expanded outward by 0.2 x 10 in both directions. Table 8 reports the performance of all considered models on ID test samples and OOD test samples. Overall, the performance measured on OOD produces increased numbers, but the trend across the compared methods remains the same. The proposed FNO,, model family produce more accurate predictions than the existing methods.
Figure 14 depeicts the per- per-parameter nRMSE measured for the reaction coefficients that are sampled in the OOD region. As the index increases (from to 10, sampled uniformly from [2.7x x 1020. 2.9x 10 the parameter is farther away from the training distribution. All considered methods show a gradual increase in nRMSE as the chosen parameter departs from the training range. The parameterized versions (pFNO,, hpFNO,) and (pFNOx, hpFNC,) achieve lower errors compared to their non-paramettrizee counterparts, indicating improved extrapolation capability. This trend expected for data-driven surrogate models, and the smooth growth of nRMSE suggests that the proposed parameterized extensions remain stable even when evaluated beyond the training domain. We note that this analysis is intended to investigate general extrapolation behavior rather than to claim superior OOD performance compared to scale or foundation models (e.g., (Rahman
al. 2024 Herde et al. 2024 Holzschuh et al. 2025
Table 8: Performance comparisons of non- pon-paramtterized (FNOc, FNO,) and parameterized (pFNOc, hpFNOc, pFNOx, hpFNOx) models for OOD samples (reaction rate). The performance is measured in the relative fe-error; the reported numerical values refer to the mean (#t standard de- viation, improvement over non- panaaaatttteeee version).
hDFVOc
pFNOx hpFNOx
NFNOc
FNOx
0.04
model performmnn mancee (PRWS))
Model
ID
OOD
FNO,, 0.0403 (#: 0.0012) 00.0628 (#: 0.0075)
FNO. 0.0375 (#: 0.0055) 0.0673 (#: 0.0097)
0.02
PFNO,
0.0334 (#: 0.0101)
0.0498 (+ 0.0069)
hpFNO, 0.0196 (::: 0.0021)
0..074 (#: 0.0051)
10
000 physical parameter
FNO, 0.0193 (::: 0.0059)
0.0046 (+ 0.0094)
PFNO, 0.0194 (+: 0.0075)
0.0343 (+ 0.0059)
Figure 14: nRMSE over the physi- cal parameter indices.
hpFNOx 0.0154 (+: 0.0029)
0.0303 (# 0.0050)
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2-dimensional parameter space Next extend our in- vestigation to 2D imput parameter space, a parameter space spanned by the reaction rate and the driving voltage. For input parameter sampling. we consider the same ranges for each pa-
Table 9: Performance comparisons
for the 2D imput parameter case.
Model
Performance
rameter Vo [100, 300), Ro [2.7 10 2.7 x 102%), con- struct uniform 2D mesh grid (21 21),,,, and randomly split them into training and test sets with 9:1 ratio. We can make observations that are similar with the ones from the ID input parameter cases; the proposed architecture FNOx provides im- provement in prediction accuracy and parameterized versions of FNO,, and FNO, further reduce the prediction errors, lead- ing to the best performance with hpFNOx, The model size and the computational timing follow the similar trends with the ID case, only except the slightly increased per-epoch computation time due to the increased number data instances the dataset for all methods.
FNOm
0.0285 (= 0.0028)
FNO-
0.0252 (= 0.0026)
CFNO
0.0223 (=: 0.0023)
MWT. 0.0305 (# 0.0028)
CMWN
0.0298 ( 0.0023)
DON: 0.0409 (# 0.0077)
U-Nete
0.0461
(+ 0.0160)
FNOx
0.0163
( 0.0009)
pFNOx hpFNO>
0.0140
( 0.0012)
0.0124 (= 0.0011)
Adaptive basis view point In the next set of experiments, vestigate further on the adaptive basis viewpoint given on the last projection operator Q. We first vary the width (that is, the number of basis functions) of the last layer of Q for and measure the prediction performance of horNOx on the vary- ing reaction rate scenario. Taking the doasis 8 as baseline, Fig-
in-
0.40
{8, 16,32 32, 64, 128}
improvement
0.30
0.25
rreeortsss the percentage of improvements (shown in the bottom
filled markers). Next, we test the idea of having (close to) orthogo-
0.15
nal basis functions and repeat the same experiments. An orthonor- mality condition imposed to the learned basis, V (a(x)); could achieve further improvement (shown in the top filled markers). As dhasis becomes small, the effect of the orthogonality enforcement becomes higher, leading to ~ 10% improvements. We define the orthogonality of the learned basis functions and its computation in Appendix B.3
16
64
128
number of basis
Figure 15: Num. of bases v.s. performance improvement.
Study on hyper-network configurations To assess the impact of hypernetwork design, we test several configurations varying in depth and width. Specifically, we consider hypernetworks with and 2 hidden layers (width 256) and hidden layer (width 128), which yield nRMSE values in the range of 0.0171-0.0179 compared to 0.0154 for the linear baseline. Deeper narrower configura- tions, such as a 3-layer network (width 256, nRMSE = 0.0200) or 3-layer network (width 32, nRMSE 0.0230), show mild degradation in performance. These results indicate that hpFNO,, perfor- mance is relatively insensitive to moderate variations in hypernetoork architecture, while deeper or overly compact designs may introduce minor instability. The linear configuration remains the most efficient and accurate choice.
Comparison of modulation techniques In addition to the proposed shift (bias) modulation, we also evaluate variants that modulate both the layer weights and biases based on the physical param- eters. We denote this approach by we'l i ght modulation. The first variant the weight modulation following the formulation suggested in the HyperFNO paper:
hw (Bty p)) W + (Uly (p)) Crow W
(17)
where hw,, (4) denotes the MLP-encoded parameter vector. This configuration allows both We(u) and se(x, (t) to vary with the physical parameter 1s.
For the next variant, we augment the additive shift modulation with multiplicative gating mecha- nism, which in similar spirit with Feature-wise linear modulation (FiLM) (Perez et al., 2018).
We denote this approach by scale modulation. With this augmentation, the Fourier layer becomes:
Uet1 (c) = a ( (4) [ ve(x)= + (K(a;; Φ) vz) + se(s,1)),
Vo D.
(18)
where the scale factor, (x, = In ....,, W)], is inferred from a hypemetwork, along with the shift u1)
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Table 10: Performance comparisons of different modulation techniques applied hpFNO,,
Reaction term
Driving voltage
Ion mass
Shift only
0.0154 (+ 0.0029)
0.0279 (+ 0.0016)
0.0128 (+ 0.0017)
Weight + shift
0.0157 (# 0.0028)
0.0214 (# 0.0056)
0.0134 (# 0.0036)
Scale + shift
0.5037 (# 0.0144)
0.0279 (#: 0.0636)
0.8550 (# 1.8748)
Scale-0.1 + shift
0.0151 (#: 0.0036)
0.0170 (#: 0.0029)
0.0137 (#: 0.0059)
Scale-0.5+ + shift
0.0146 (# 0.0022)
0.0180 (# 0.0031)
0.0157 (# 0.0053)
Compared to using shift term alone, the inclusion of ye(2,, (4) makes the modulation strictly more expressive, as it enables the network to multiplicatively amplify or suppress the latent responses in parameter-dependent manner. However, multiplicative gating can introduce training instabilities if "e varies too aggressively across the parameter space. To mitigate this, we employ a stabilization strategy by parameterizing the gate as
(e. n1) tannh(Se
(19)
which keeps the modulation centered near identity at initialization while still allowing smooth, learn- able deviations driven by the physical parameter. Here, 7 becomes the hypemeework output and is the hyper-parameter to decide the amplitude of deviation. We denote this variant by scale-1, where the suffix indicates the chosen value of 17.
Table provides summarized results. Overall, the amplitude-basee multiplicative gating generally offers performance improvements, but it introduces several considerations. First, naive multiplica- tive modulation is unstable, since the unconstrained scaling factors can deviate excessively and cause large errors during training. Second, the stabilized formulation proposed in this work, which applies scaled tanh function control the amplitude of the gate, significantly improves robustness but requires searching over the hyperparameter 1) to determine the appropriate scaling range. Third, the benefits of multiplicative gating are not universal across all settings; for example, in the ion mass benchmark, the configuration with the shift modulation alone achieves the best accuracy. These observations indicate that although multiplicative gating is more expressive in principle and can yield improved performance, its practical effectiveness depends on careful amplitude control and characteristics of the specific parameterized problems.
In summary, reasonably strong and stable parameterization can be obtained using the shift mod- ulation alone, which delivers reliable performance across the benchmarks considered in this work. However, the amplitude based multiplicative gating provides additional expressive capacity, and with a more refined search over its scaling hyperparameters, there is room for achieving higher ac- curacy. This suggests practical tradeoff: the shift modulation is simple and robust, whereas the amplitude based modulation can offer further performance gains when carefully tuned.
E GRAY-SCOTT EQUATIONS
The Gray-Scott equations form system of coupled reaction-diffusion equations that describe the spatiotemporal dynamics of interacting chemical species. This model can reproduce a rich variety of natural patterns, including structure resembling bacterial colonies, spiral waves, and coral-like formations. In the system, two state variables and v each undergo independent diffusion and linear growth/decay, while their interaction is governed by the nonlinear reaction term.
The governing equations are defined follows:
Tu = él Ox
82u
+ F(1 - u) - >uuv?,
01
(20)
Ou = 0x (K + F)u h + 2zue?
Øt
where u(z t) denotes concentration of the feed species, which is continuously supplied to the system and v(x, t) denotes concentration of the reactant species, which is produced and consumed through interaction with The diffusion coefficients of and are denoted as e and 62, respectively. Other
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parameters include: F, the feed rate, representing the rate which species u is introduced into the system from an external source, K, he decay rate of species v, which accounts for natural removal, and X, the reaction rate constant associated with the nonlinear term -Aw??, which describes the autocatalytic process where one unit of u reacts with two units of v.
E. EXPERIMENTATION
In the experiment, consider the parameterized dynamics realized by varying the diffusion co- efficient é E [0.1, 10j and varying the feed rate F E [0.1, 10). We collect 101 samples of equi-distanced coefficient/feed rate values while the other parameters are fixed as e2 0.05, K = 0.1 and (A1, 22) = ().5). For sampling an initial condition, we follow the approach described in (Xiao et al. 2023). sampling u(z,0) 0) using the smooth random functions in the Chebfun package (Drscooll et al. 2014) and sampling v(z, 0) from Gaussian random field, v(x, 0) NO 7-2x (-2-- - 7 (0))) For collecting trajectories for constructing training/test datasets, initial value problems are numerically solved using fourth-order time-integrator, expo- nential time differencing fourth-order Runge- Kutta (ETDRK4), utilizing the chebfun software. We base the implementation the publicly available Mat lab scrip which is the implementaion accom- panyed by the research work (Xiao et al. (023). The IVPs are solved with a spatial resolution, 1024, and then the collected solution snapshots subsampled in the spatial domain, leading to the final spatial resolution, 128. The time integration is performed for the time interval [0,0.1] with 31 time steps.
Table 11: Performance comparisons between the proposed methods and the baseline methods. All models are tested scalar parameter: the diffusion coefficient. The performance is measured in the relative eerrrror the reported numerical values refer to the mean (= standard deviation)
Model
Diffusion coeff,
Feed rate, F
FNOm
0.0129 (= 0.0032)
0.0116 (+ 0.0026)
FNO.
0.0089 (# 0.0052)
0.0097 (#: 0.0030)
CFNO
0.0093 (#: 0.0016)
0.0161 (:: 0.0045)
MWT-
0.0055 (= 0,0011)
0.0092 (# 0.0021)
CWMNO
0.0048 (+: 0.0015)
0.0293 (-t 0.0078)
D
0.0243 (= 0.0029)
0.0138 (= 0.0020)
U-Net
0.0239 (+ 0.0160)
0.0159 (+ 0.0071)
FNOx
0.0039 (= 0.0002)
0.0075 (= 0.0016)
pFNOx
0.0027 (= 0,0008)
0.0041 (= 0.0010)
hpFNOx
0.0022 += 0.0010)
0.0022 (= 0.0006)
Prediction accuracy Table [1 reports the prediction accuracy of all considered method. The re- sults provide similar observations with those of the ID CCP experimentation: the proposed methods, the parameterized variants of FNO, achieve the lowest errors compared to existing baselines with 54% (for the diffusion coefficient case) and 72% (for the feed rate case) improvements. The coupled extension of FNO, FNO,, improves the performance compared to other basclines. This is achieved without significantly scarifying the training/inference time and the memory footprint compared to FNCm or FNO, while other baselines require significantly larger-sized models and/or computational times. In the varying feed rate scenario, the approaches leveraging two separate operators struggle further to capture the accurate dynamics, resulting in higher prediction errors.
Computational/memory aspects Figure 16 further provides more information on computing. The figure presents the model sizes (i.e., memory requirements) and the computational timing for per epoch operation measured in seconds. The first row presents the results for the diffusion coeff- cients and the bottom row presents the results for the feed rate. Again the proposed variants pFNO, and hoFNO, serve as an efficient (in both memory and computation) and effective surrogate for modeling the parameterized Gray-Scott equations. Along with these information, the third column
*This script is available http://ghhiaccmm/ssssriiiokkoooo
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of Figure 16 presents the test loss achieved over the course of the training. The proposed hpFNO, demonstrates faster convergence to more accurate predictions.
FMOm
CFNO
pFNOX
MWT CHINNO
DONc
PMX
hprNOx
0.204
6,075
10"
0.020
(nitSS))
0.020
model performent (n0044))
model perfrmonce
0.015
0.015
20n1
0.010
0.010
10
0.05
30
0.58
142
of model 1x10%)
Eminng Isec)
Epoch
(a) Model size for 61
(b) Timing for 61
(c) Test loss for 61
0.030
(WROSSE)
model netforment (VRONE))
perfomancc
0.020
0.020
0.015
0.015
0.010
0.010
0.005
0.005
0.58
142
1.00
200
500
400
500
number of model parameters tx20")
timing
(d) Model size for
(e) Timing for F
(() Test loss for F
Figure 16: [GS] The results of the experiments for varying diffusion coefficients (top) and varying feed rates (bottom). Plots depict the number of model parameters the model performance (left), computational timing versus the model performance (middle), and the test loss trajectories over epochs (right).
Varying spatial resolutions We further conduct additional exper- iments on the Gray-Scott benchmark of varying diffusion coeffi- cients using datasets generated at three different spatial resolutions: 64, 128, and 256 grid points. All models are trained separately on data from each resolution under identical training configurations and evaluated on corresponding test sets. The results, summarized in Figure 17 show that all considered methods maintain compa- rable accuracy across resolutions, consistent with the resolution- invariant nature of neural operator formulations. Among them, FNCx consistently achieves the lowest relative (2-errors (approx- imately 3.3 10 across all grid sizes). These results confirm that the proposed architecture preserves accuracy under varying spatial discretizations while maintaining the efficiency of operator learning.
ONO
320
256
Figure 17: [GS] aRMSE over varying resolutions
F 2D CHEMICALLY REACTING FLOW
As next benchmark, we consider the two-dimensional chemically reacting flow of premixed H2- air flame at constant and uniform pressure Buffoni Willcox (201)). which is modeled as a coupled system of four PDEs. Each equation governs the temporal and spatial evolution of one of the four thermo-chemical state variables, temperature and the mass fractions H,, O2, and H,O, which are nonlinearly coupled through diffusion, convection, and reaction terms. The governing equations for
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the system is expressed
Ów(axt;
tx - - Vw(wx + ((w(x, t; p));
(21)
Ōt
where w = [WT,, WH2; wO2, WH represents the state vector of the four coupled fields.
On the right-hand side, the first term captures molecular diffusion with diffusivity 2 cm² the second term represents advection by a constant, divergence-free velocity field = [50 em/s, 0] and the third term q accounts for nonlinear chemical reactions coupling all species. The system is therefore strongly coupled: the evolution of each variable depends on the others through and, to lesser degree, through diffusive transport.
The reaction source term follows an Arthenius-type law:
q ww: /1) = [gr (w: po), qH, (w; pt), qo. (w: μ)) ( p)]
(22)
where
gr(w: μ)) = Q (w:
(:w; (L) -t
W:
pWH2
pwoz
v02
Ae
NeT
(23)
p
WH.
Wo
for E (H2, H2O).
The stoichiometric coefficients are (UH2r t.2, UH2O) 1, -2,, and the molecular weights
(WH,, Wo,, WH, = (2.016, 31.9, 18) g/mol. The mixture density p = 1.39 x 10-3 g/em², the universal gas constant is R = 8.314 J/(mol-K), and the heat of reaction is = 9800 K.
The system depends on two physical parameters, μ (2) = E), corresponding to the pre- exponential factor and activation energy, respectively. Hence, this problem represents four-variable coupled nonlinear PDE system with two input parameters, capturing the essential thermo-chemical coupling of a reacting Hy-air mixture.
Figure 18 illustrates the computational geometry and the cor- responding boundary conditions, defined as follows:
31
mm
T2: inflow boundary with Dirichlet conditions spec- ified by
mm
T1 (wall)
Fg
950 K and (WHa, Wos, WHoo) (0.0282, 0.2259, 0),
mm
(inlet)
T3
I and 33: boundaries with Dirichlet conditions T 300 K and (WH2r Wo,,, WH₂0) (0, 0, 0),
(wall)
1'4. I's, and rg:: boundaries with homogeneous Neu- mann conditions,
Figure 18: [ADR] The spatial do-
main and boundary conditions.
and the initial condition is (WHz) Wo,; WH₂O) = (0, 0, 0), indicating that the domain initially contains no chemical species.
T 300 K and
F.1 EXPERIMENTATION
The numerical data are generated using difie-difference scheme on a uniform grid of 64 x2 points (i.e.. 4 x 64 x 32, 4 variables). For training and testing NOs, we subsample the horizontal direction to 32, leading to 32 x 32 for four variables. Time integration employs the second-order backward differentiation formula (BDF2) with constant time step Al = 10-1s s and final simulation time of 0.02 (corresponding to n, = 200 time steps). For training and testing NOs, we subsample every 5th snapshot, resulting in 40 temporal indices for each trajectory.
The two model parameters, the pre-exponential factor A and the activation energy E, are sampled uniformly from the rectangular parameter domain A, E) E [2.3375 10 6.5 10 x [5.625 x 103, 9 x 10 For data collection, we construct an 8 8 uniform grid over this domain, resulting in 64 trajectories, which are then split into 56:8 (train:test).
Figure 19 summarizes the performance and efficiency of the proposed and baseline models. this experiment, consider subset of the haselines considered in other experiments, namely FNO.,
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cWTc, and DONc. Particularly, CFNO and CMWNO excluded from this comparison, their chitectures are primarily demonstrated for two-variable systems and it is hard to find their extension to higher-order coupling is not explicitly discussed in the original formulations. In the results, observe that the proposed variants, FNOx and hpFNO,, achieve the lowest prediction errors while maintaining competitive model size and computational cost. Relative FNO., FNOx improves curacy by about 41%, and hpFNOx further reduces the error by approximately 61 %. Compared to the multiwavelet baseline MWT., hpFNOx still provides around 41% lower error while remaining lightweight. DON, yields nRMSE values roughly order of magnitude larger than the other mod- els and therefore omitted from Figure for visual clarity. Overall, the results demonstrate that spectral coupling and parameter conditioning enhance predictive accuracy and efficiency without compromising the compact structure of the original FNO architecture.
FhOc
FNOc
FNCC
perfoormmette (nIMSS) (410-)) paett
FlOX
modee peroomence (nDaQt) bxit"))
FHOx hpFhOX
RNCx
hcF
hoFhOx
performance (nRMSS))
10.1
mocoi
10:2-
1.5
8.5
100
200
500
400
500
number model parameters (x10")
timing tsec)
(a) Model size
(b) Timing
(c) Test loss
Figure 19: [ADR] The results the experiments for varying pre-exponential factor and the acti- vation energy (4, E). Plots depict the number of model parameters versus the model performance (left), computational timing versus the model performance (middle), and the test loss trajectories over epochs (right).
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Reviews:
[{"review_id": "Qi2sgU0NdV", "reviewer": "Reviewer_tkqC", "summary": "1. The paper is well-written, easy to follow, and clearly motivated.\n\n2. The experimental results are impressive, demonstrating a 55-72% reduction in error over strong baselines and confirming the value of the principled design.\n\n3. I am not an expert in this area and will defer to the other reviewers for their expert opinions on the technical contribution", "questions": "What is the key difference between your method and the baselines, and why does your approach work?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written, easy to follow, and clearly motivated.\n\n2. The experimental results are impressive, demonstrating a 55-72% reduction in error over strong baselines and confirming the value of the principled design.", "weaknesses": "The manuscript would benefit from additional visualizations, such as a diagram illustrating the model architecture, to help readers better understand the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aRzY4AhHmq", "reviewer": "Reviewer_FkHd", "summary": "This paper extends the Fourier Neural Operators (FNOs) to handle coupled and parameterized PDEs by introducing new architectures. The authors propose designs that model parametrized dynamics and coupled systems and evaluate it on a newly developed 1D capacitively coupled plasma and Gray-Scott dataset. Experimental results demonstrate that the proposed hpFNO models achieve higher predictive accuracy compared to existing FNO variants.", "questions": "1. What exactly does “acc” in Figure 3 represent? It seems the lower is better, so 'acc' might be misleading.\n2. How many CCP benchmark data were generated, and how were they split into training/test sets?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "* The introduction clearly motivates the need for efficient operator learning on coupled and parametrized PDEs.\n* The proposed pFNO, and hpFNO architectures aim to improve representations for coupled and parametrized PDEs, and the results on the benchmarks are promising.\n* The ablation study thoughtfully analyzes the effects of design components, showing careful empirical work.", "weaknesses": "1. Lack of architectural visualization : The paper introduces multiple new and extended operators, but there is no schematic figure summarizing the overall architecture.\n2. Unclear explanation for “best 5 of 10 runs” in Table 2 : Why is this criterion used for $T_{in} = 2$ and $T_{in} = 1$?\n3. Sparse content in Section 4.3 : This section contains only a brief summary and refers the reader to the appendix. Including a compact summary table would be better for understanding cross-domain generalization.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5odrjFTdgm", "reviewer": "Reviewer_6Lqg", "summary": "This paper extends Fourier Neural Operators (FNOs) to handle parameterized and coupled PDEs. The authors propose two lightweight modifications: (1) conditioning FNOs on physical parameters through either input concatenation (pFNO) or a small hypernetwork modulation (hpFNO), and (2) enabling coupling between multiple PDE variables by performing cross-variable mixing in Fourier space. The resulting framework (FNOx, pFNOx, hpFNOx) retains the efficiency of standard FNOs while improving accuracy for systems with varying parameters or interacting fields. The authors introduce a new benchmark based on a 1-D capacitively coupled plasma model and also evaluate on the Gray–Scott equations. The proposed hpFNO achieves up to 55–72 % error reduction over strong baselines.", "questions": "1. How sensitive is the hpFNO performance to the design of the hypernetwork (depth, parameterization, modulation type)?\n\n2. Could the same effect be achieved by allowing \\( W_\\ell(\\mu) \\) (parameter-dependent local maps) instead of additive \\( s_\\ell(x, \\mu) \\) biases?  \n\n3. Are there stability or extrapolation limitations when \\( \\mu \\) lies outside the training range?  \n\n4. How does coupling only in Fourier space compare to coupling in both spatial and spectral domains?\n\n5. Can the authors comment on scalability to 2-D/3-D PDEs and memory growth with the number of coupled variables?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Addresses a timely and important gap: FNOs are increasingly popular for PDE surrogates, yet parameterization and coupling remain under-explored.\n\n- The architecture modifications are simple and principled—hypernetwork modulation and Fourier-space coupling can be integrated into existing FNOs with minimal effort.\n\n- The approach is general and modular, applicable to multidimensional parameter fields and different neural-operator families.\n\n- Demonstrates comparisons across multiple strong baselines (FNO, CFNO, MWT, DeepONet, U-Net) with solid quantitative gains.\n\n- Introduces a new plasma-physics benchmark that could be useful to the community.", "weaknesses": "1. Writing is meandering and often unfocused; core ideas and notation could be stated more directly.\n\n2. Organization: the Related-Work section appears late; it would be clearer to position it before the Methods.\n\n3. The Gray–Scott example feels secondary, lacking detailed analysis or ablation—appears appended to strengthen results. In fact, the experiments are limited and cannot demonstrate general applicability. The paper should be enhanced with more experiments from different PDEs.\n\n4. The claimed generality for “parameterized and coupled PDEs” is convincing for 1-D cases but not well demonstrated on higher-dimensional or more complex systems.\n\n5. Limited qualitative insight—plots focus on error metrics. The authors should delve deeper into the interpretation of learned coupling or modulation effects.\n\n6. typos like \"learable\". should have numbered the equations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ea4uvIcM87", "reviewer": "Reviewer_UVnJ", "summary": "This paper proposes two architectural extensions to Fourier Neural Operators (FNOs) to better handle two important classes of PDEs: (i) parameterized dynamics with fixed initial conditions, and (ii) systems of coupled PDEs with interacting physical fields. For parameterized dynamics, the authors introduce a lightweight hypernetwork that modulates FNO layers based on physical parameters. For coupled PDEs, they perform a systematic architectural design study to determine where and how to introduce cross-variable coupling, ultimately proposing a Fourier-space interaction scheme. The final model, FNOx (and its parametric variants pFNOx, hpFNOx), achieves significant gains over strong baselines — including coupled multiwavelet neural operators — on both a newly proposed 1D capacitively coupled plasma benchmark and the standard Gray–Scott system.", "questions": "1. **How does the hypernetwork-modulated FNO behave on out-of-distribution (OOD) physical parameters?** Would performance degrade sharply or smoothly? Have you tested its extrapolation limits?\n\n2. **Why is the coupling introduced only in the Fourier domain?** Have you explored combining it with local (real-space) cross-variable interactions, and how would that affect performance or efficiency?\n\n3. **Can your proposed architecture scale to PDE systems with more than two coupled fields?** If not, what limitations would arise — architectural, memory, or training-related?\n\n4. **How would the design choices in FNOx (e.g., Q2c, L2, G) transfer to more challenging PDE domains like turbulent 2D/3D fluid flow or real-world inverse problems?** Do you expect the same configurations to hold?\n\n5. **To what extent does the performance improvement come from architectural tuning versus true modeling of coupled/parameterized dynamics?** Could stronger baselines (e.g. HyperFNO, CoDA-NO) close the gap?\n\n6. **In the case of high-dimensional parameter spaces (e.g., 10–20 physical parameters), would the hypernetwork remain effective without overfitting or becoming too large?** How would you scale it?\n\n7. **How sensitive is the performance to the size and architecture of the hypernetwork?** Could a different modulation scheme (e.g., multiplicative gating) outperform simple bias shifts?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* **Addresses underexplored settings**: The paper targets two highly practical yet understudied scenarios in neural operator learning — parameterized dynamics with fixed initial conditions, and multiple interacting PDE fields. Both are common in engineering and physics simulations.\n\n* **Effective, minimal extensions**: The proposed improvements are simple but well-motivated. The hypernetwork-based modulation (hpFNOx) conditions model behavior across parameters without blowing up model size, and the spectral-domain coupling design elegantly leverages FNO’s strengths.\n\n* **Systematic architectural study**: The paper carefully explores different configurations (shared vs separate layers, coupling placements) and justifies its final FNOx architecture based on empirical and structural considerations.\n\n* **Strong empirical results**: On both benchmarks, the proposed models achieve large performance gains — often reducing error by 50–70% relative to the best baseline — while maintaining similar runtime and parameter count.\n\n* **New benchmark dataset**: The 1D plasma physics setup with tunable parameters offers a meaningful and realistic testbed for studying parametric and coupled operator learning.\n\n* **Clarity and reproducibility**: The paper is generally well-written, with detailed experimental setups and promised code release. It adheres to good reproducibility practices.", "weaknesses": "* **Incremental novelty**: The core ideas — hypernetwork modulation and coupling in neural operators — are grounded in existing techniques. HyperFNO and related works already explored hypernetwork-based parameter conditioning, and Fourier-space coupling is a natural extension within FNOs.\n\n* **Narrow scope**: The benchmarks, while appropriate, are relatively small-scale and low-dimensional (1D/2D). It’s unclear how well the approach generalizes to larger or more complex systems (e.g. 3D Navier–Stokes, multiple coupled fields).\n\n* **Limited analysis of generalization**: There’s little discussion on how well hpFNOx handles out-of-distribution parameters or extrapolation, which is a practical concern in real applications of parameterized PDE solvers.\n\n* **Brief treatment of second benchmark**: The Gray–Scott results are reported only briefly in the main paper, which weakens the case for generality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)