Helix-FNO: Spectral-Domain Operator Learning Coupled with a High-Fidelity Mechanistic Model for Fast Surrogate Simulation

Jiabao Zhao; Chuwei Wang; Jinxi Yang

cs.LGAug 23, 2026

Helix-FNO: Spectral-Domain Operator Learning Coupled with a High-Fidelity Mechanistic Model for Fast Surrogate Simulation

Authors: Jiabao Zhao, Chuwei Wang, Jinxi Yang

Organizations: University of Science and Technology Liaoning Anshan, Liaoning, China · Wuhan University Wuhan, Hubei, China

Abstract

Mechanistic simulation models of full-scale treatment processes remain the only trustworthy, extrapolative description of the underlying physico-chemical dynamics, yet their runtime is far too slow to support the thousands of forward evaluations that a modern decision engine requires at a 5-minute decision cadence. The standard remedy-surrogate modelling-often produces a network that learns a single solution for a single configuration, so it generalises poorly to new influent profiles, control settings or plant layouts. This paper presents Helix-FNO, a teacher-student architecture that couples a thirty-two-state mechanistic teacher with a Fourier neural operator (FNO) student. The teacher supplies a high-fidelity dataset of input-field-to-solution pairs, curated by Latin-hypercube and uncertainty-based active learning to cover the boundary and overload regimes that matter in practice; the student learns, in the spectral domain, the solution operator itself rather than any single solution, thereby moving from learning one instance to learning an entire family of equations. We give the operator formulation, the spectral convolution definition, the weighted distillation loss and the active-learning criterion, and we analyse the approximation error of a truncated Fourier expansion with respect to the smoothness of the parametric solution manifold. An illustrative study compares Helix-FNO against a physics-informed network and a data-driven recurrent surrogate on accuracy, dataset efficiency and inference latency, and places the methods on a speed-accuracy Pareto front. The resulting operator is three orders of magnitude faster than the mechanistic teacher at millisecond inference, which is precisely the capability required for massive candidate screening and online decision support.

Figures & tables

Figure 1: The Helix-FNO teacher–student architecture. The 32-state mechanistic teacher ( Helix-Core ) generates a curated high-fidelity dataset of input-field–solution pairs (LHS + active learning); the FNO student lifts, applies $L$ spectral convolution layers and projects to predict $\hat{u}(x,t)$ , and is trained by a weighted distillation loss against the teacher output.

Figure 2: Illustrative spectral energy distribution of the parametric solution operator. The per-mode energy (bars) decays sharply with wavenumber $k$ , so the cumulative energy (curve) saturates rapidly; the truncation $K_{\max}=16$ retains almost all energy while filtering the high-frequency noise floor.

Component	FNO student	PINN (per-instance)	LSTM (data-driven)
Lifting dim $d_{v}$	$64$	–	–
Spectral layers $L$	$4$	–	–
Mode truncation $K_{\max}$	$16$	–	–
Hidden size	–	$64\times 4$	$128$
Depth / layers	$4$	$4$	$2$
Activation	GELU	$\tanh$	$\tanh$

Table 1: Illustrative model configuration and hyper-parameters.

Figure 3: Illustrative accuracy versus teacher dataset size. The FNO operator (blue) reaches the lowest relative RMSE and saturates the fastest, whereas the per-instance PINN (red) and the data-driven LSTM (green) remain higher at the same sample budget.

Setting	COD	NH 3 -N	TN	TP
In-distribution (plant A)	$2.8\%$	$3.1\%$	$3.5\%$	$4.2\%$
Zero-shot (plant A $\to$ B)	$7.9\%$	$8.6\%$	$9.4\%$	$11.2\%$
Few-shot (32 samples)	$3.6\%$	$4.0\%$	$4.5\%$	$5.3\%$
Fine-tuned (256 samples)	$2.9\%$	$3.2\%$	$3.6\%$	$4.4\%$

Table 2: Illustrative relative RMSE for cross-plant transfer and zero-shot inference, after domain-adaptive normalisation. Values are illustrative.

Figure 4: Illustrative inference-speed vs accuracy Pareto front. The mechanistic teacher (bottom-right) is the most accurate but slowest; the FNO operator (centre-left) attains near-teacher accuracy at millisecond latency, dominating the per-instance PINN and offering the best trade-off against the fast but less accurate LSTM.

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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Jun 9, 2026

SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators

Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately

4

15

times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of

73

times fewer parameters across multiple PDE benchmarks.

Pengqing Shi, Jie Yin, Stephen Tierney +1

The University of Sydney, Australia

Fourier Neural Operator Spectral Bias PDE Operator Learning Sinusoidal Representation Networks

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May 27, 2026

Sequential Physics-Constrained Neural Operator Forward Modeling for the $\textit{Norne}$ Reservoir System

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous

46\times112\times22

grid (

N=113,344

cells), with a production history spanning

T=30

timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as

W_2 \leq \varepsilon(L^n-1)/(L-1)

, with exponential population-risk discrepancy for

L>1

; (3) physics-constrained spectral stability, showing PINO training with

λ_R \geq λ^*_R

reduces the learned Jacobian spectral radius to

ρ_F + Cλ_R^{-1/2}

, yielding uniform-in-time rollout error

|δ_n| \leq \varepsilon/(1-ρ)

; and (4)

K

-step TBPTT gradient analysis, deriving geometric bias decay

O(ρ^K)

, optimal window

K^ = O(\log(T/σ^2))

, and Adam convergence

O(1/\sqrt{t}) + O(ρ^{K^*})

. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain

R^2>0.99

(oil),

R^2>0.90

(gas),

R^2\approx 0.80

(pressure), and monotonically improving

R^2

(water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a

{\sim}10^4\times

wall-clock speedup over the OPM finite-volume simulator.

Clement Etienam, Juntao Yang, Oleg Ovcharenko +4

NVIDIA Corporation

Fourier Neural Operator PDE Surrogate Modeling Physics-Informed Neural Operators Computational Fluid Dynamics

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May 10, 2026

HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field

u(t,\cdot)

is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time

t

and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state

u_t(θ,x)=u(t+θ,x)

θ\in[-τ,0]

. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from

0.241

0.188

, and

0.185

for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to

0.094

. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

Lennon J. Shikhman

College of Computing, Georgia Institute of Technology Atlanta, GA, United States

Fourier Neural Operator PDE Solving PDE Surrogate Modeling PDE Operator Learning Neural Operators

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Abstract

Neural Surrogate Modeling

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