Mirror-Padded Fourier Neural Operator (MFNO)

• The paper introduces MFNO, which extends classical Fourier neural operators by incorporating mirror padding to handle non-periodic stochastic inputs effectively.
• MFNO employs a boundary treatment that creates a smooth periodic extension, ensuring compatibility with Sobolev-space approximation theory and enabling operator learning for path-dependent SDEs and fBM.
• Empirical results demonstrate that MFNO achieves competitive accuracy, strong resolution generalization, and faster sample-path generation compared to traditional Euler and vanilla FNO methods.

Searching arXiv for the MFNO paper and closely related FNO context. The mirror-padded Fourier neural operator (MFNO) is an operator-based neural network designed to learn the dynamics of stochastic systems, particularly path-dependent stochastic differential equations (SDEs) and processes driven by fractional Brownian motion (fBM). It extends the standard Fourier neural operator (FNO) by incorporating mirror padding, thereby enabling the Fourier-operator framework to handle non-periodic inputs on a finite interval while retaining global convolutional structure. In the formulation introduced in 2025, MFNO is presented as a mathematically compatible adaptation of FNO to non-Markovian stochastic dynamics, together with approximation theorems and experiments emphasizing resolution generalization, competitive accuracy, and fast sample-path generation (Lee et al., 23 Jul 2025).

1. Conceptual setting and problem class

MFNO is motivated by a structural mismatch between standard FNOs and stochastic time-series data. Ordinary FNOs are naturally defined for periodic functions on the torus $\mathbb{T}^d$, whereas sample paths of Brownian motion, path-dependent SDE solutions, and fBMs on a finite interval $[0,T]$ are not periodic. The central purpose of MFNO is to preserve the spectral and operator-learning advantages of FNO while adapting the architecture to such non-periodic path data (Lee et al., 23 Jul 2025).

In this setting, the target objects are operators on paths rather than static finite-dimensional maps. For path-dependent SDEs, the future evolution depends on the entire past trajectory, so the relevant learning problem is naturally framed as approximation of a solution operator from an initial condition and a driving path to the resulting solution path. The same operator perspective appears for fBM-related tasks, where Brownian input is transformed into a rougher or smoother path through an integral kernel, and a subsequent operator on function space is learned.

This operator viewpoint distinguishes MFNO from sequence models that are tied to a fixed discretization. A plausible implication is that the architecture is designed not merely for next-step prediction but for mesh-aware functional approximation across resolutions. That implication aligns with the reported emphasis on resolution generalization, which is treated as a defining empirical property of the model.

2. Architectural definition and mirror-padding mechanism

The standard FNO is written as

$$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$

where $\mathcal{R}$ lifts the input into a latent feature space, $\mathcal{L}_\ell$ are nonlinear Fourier layers, and $\mathcal{Q}$ projects back to the output space. Each Fourier layer is given by

$\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$

with Fourier weights $P_{\theta_l}(k)$ parameterizing the kernel in frequency space. A discretized variant, termed the $\Psi$-FNO, inserts pseudo-spectral projections $\mathcal{I}_N$ between layers: $[0,T]$0

MFNO augments this construction by inserting a mirror-padding layer $[0,T]$1 before the FNO and a truncation layer $[0,T]$2 afterward. In one dimension, the mirror-padding map is

$[0,T]$3

and the resulting network is

$[0,T]$4

The mirror-padding operation produces a continuous, periodic extension on $[0,T]$5. After Fourier processing on the doubled interval, the truncation layer restricts the output back to the original time domain. The architectural novelty is therefore not a change to the internal Fourier layer formula itself, but the addition of a boundary treatment that makes non-periodic paths compatible with the periodic assumptions underlying the FNO framework.

3. Functional-analytic rationale for mirror padding

The justification for mirror padding is explicitly Sobolev-theoretic. Standard FNO approximation theory is formulated on periodic domains, and the universal approximation theorem recalled in the paper concerns continuous operators between Sobolev spaces on compact sets. For non-periodic stochastic paths on $[0,T]$6, a naive extension can violate the regularity needed for that theory (Lee et al., 23 Jul 2025).

The contrast with zero padding is central. If one zero-pads a non-periodic path, the extended function is typically discontinuous at $[0,T]$7, and may therefore fail to belong to $[0,T]$8. Mirror padding avoids this artificial jump by reflecting the signal, thereby creating an extension that is continuous and periodic on the doubled interval. This is the mechanism by which the input becomes smooth enough for the Sobolev-based FNO theory to apply.

This functional-analytic point is not merely cosmetic. In the paper’s presentation, mirror padding is the device that reconciles two otherwise incompatible objects: Fourier layers formulated on periodic domains, and stochastic sample paths observed on a finite interval. A plausible implication is that MFNO’s boundary treatment is part of the approximation theorem itself, rather than an implementation detail added only for empirical convenience.

4. Approximation theory for path-dependent SDEs and fractional Brownian motion

For path-dependent SDEs, the paper considers equations of the form

$[0,T]$9

with $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$0 and $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$1 non-anticipative functionals on path space. The associated solution operator is

$$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$2

Because this operator is generally only measurable and not continuous, the analysis proceeds through a Wong--Zakai approximation. Brownian motion $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$3 is replaced by its non-adapted piecewise linear interpolation $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$4 on a partition $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$5, and one studies

$$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$6

where $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$7 is the Stratonovich-corrected drift. The key estimate is

$$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$8

This estimate allows the stochastic system to be approximated by a deterministic equation driven by a smooth path. Once this regularization is in place, the solution operator becomes continuous on a compact subset of $$\mathcal{N}=\mathcal{Q}\circ\mathcal{L}_{L}\circ\mathcal{L}_{L-1}\cdots\circ\mathcal{L}_{1}\circ\mathcal{R},$$9, and MFNO’s universal approximation theorem can be invoked. The resulting theorem states that for any $\mathcal{R}$0 and any finite range of interpolation levels, there exist $\mathcal{R}$1, $\mathcal{R}$2, a high-probability event $\mathcal{R}$3 with $\mathcal{R}$4, and an MFNO $\mathcal{R}$5 such that

$\mathcal{R}$6

for $\mathcal{R}$7 (Lee et al., 23 Jul 2025).

For fractional Brownian motion, the covariance is recalled as

$\mathcal{R}$8

together with the stochastic integral representation

$\mathcal{R}$9

The paper defines

$\mathcal{L}_\ell$0

and shows that $\mathcal{L}_\ell$1 is continuous. It also uses the approximation

$\mathcal{L}_\ell$2

and the fact that $\mathcal{L}_\ell$3 converges to $\mathcal{L}_\ell$4 in $\mathcal{L}_\ell$5.

The corresponding theorem asserts that for any $\mathcal{L}_\ell$6-Lipschitz operator $\mathcal{L}_\ell$7, there exist $\mathcal{L}_\ell$8, and an MFNO $\mathcal{L}_\ell$9 such that

$\mathcal{Q}$0

for sufficiently fine interpolations $\mathcal{Q}$1. A corollary treats the special case $\mathcal{Q}$2, showing that MFNO can approximate fBM itself from Brownian input.

Across both analyses, the proof strategy has the same structure: regularize the stochastic input, prove continuity of the induced deterministic operator on $\mathcal{Q}$3, apply MFNO universal approximation on compact Sobolev subsets, and control the stochastic approximation error separately. The role of Wong--Zakai type theorems is therefore foundational rather than auxiliary.

5. Empirical behavior, baselines, and computational profile

The empirical study trains MFNOs on two path-dependent SDEs and on fBM with Hurst parameters $\mathcal{Q}$4 and $\mathcal{Q}$5. For the SDEs, training data are generated by Euler simulation, and evaluation is reported on resolutions ranging from $\mathcal{Q}$6 to $\mathcal{Q}$7. The principal empirical result is strong resolution generalization: when trained at one grid resolution, MFNO can produce accurate outputs on finer grids without retraining, and its error remains nearly stable as resolution increases (Lee et al., 23 Jul 2025).

This behavior is contrasted with vanilla FNO, whose performance degrades as resolution increases, reflecting boundary mismatch and wrap-around artifacts. Zero-padded FNO (ZFNO) also achieves good resolution generalization, but MFNO is presented as more theoretically principled because zero padding introduces a discontinuity at the boundary. On the resolution-$\mathcal{Q}$8 tests, MFNO is reported as highly competitive and often best or near-best in relative $\mathcal{Q}$9 and $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$0 error for the path-dependent SDEs; on the second SDE it attains the lowest reported $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$1 error among all baselines.

The baseline set comprises vanilla FNO, ZFNO, DeepONet, TCN, and LSTM. DeepONet is generally less accurate than the FNO-based methods. TCN and LSTM can fit fixed-resolution data, but they are not naturally resolution-invariant, so they are not evaluated on the variable-resolution experiments. For fBM, the results depend on roughness: on the smoother case $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$2, FNO-family models perform well, whereas on the rougher case $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$3, all models struggle more.

The computational comparison emphasizes sample-path generation speed. Euler simulation is reported at about $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$4 ms per sample, whereas MFNO takes about $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$5 ms, with FNO and ZFNO being slightly faster still. The stated complexity comparison is $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$6 for Euler-type methods versus $\mathcal{L}_{l}(v)(x)=\sigma\left(W_{l}v(x)+b_l(x)+\mathcal{F}^{-1}\left(P_{\theta_{l}\cdot \mathcal{F}(v)\right)(x)\right),$7 for FNO-style FFT-based evaluation. This suggests that MFNO’s practical value lies not only in operator approximation and cross-resolution behavior, but also in amortized generation speed once training is complete.

6. Relation to standard FNO, zero-padded variants, and non-MFNO architectures

MFNO should be situated within the broader family of Fourier neural operators, but it is not interchangeable with all recent FNO variants. Its defining modification is boundary-aware preprocessing for non-periodic path data through mirror padding, followed by truncation after Fourier processing. The core problem it addresses is the incompatibility between periodic spectral architectures and non-periodic stochastic inputs on finite intervals (Lee et al., 23 Jul 2025).

This distinguishes MFNO from zero-padded FNO. Both approaches extend non-periodic data to a larger domain before Fourier processing, and both are reported to generalize well across resolutions. The difference is theoretical: zero padding introduces a discontinuity at the boundary, whereas mirror padding avoids the artificial jump and preserves access to Sobolev-space approximation arguments. The distinction is therefore not merely empirical but analytic.

MFNO is also distinct from FNO variants that target difficult PDEs through architectural changes unrelated to boundary extension. A notable example is the 2023 model titled "Enhancing Solutions for Complex PDEs: Introducing Complementary Convolution and Equivariant Attention in Fourier Neural Operators," which proposes a hierarchical Fourier neural operator with convolution-residual layers and large kernel attention to address multiscale PDEs. That method combines global spectral processing with local convolution and attention, but it does not describe mirror padding, reflection padding, or an MFNO-style boundary treatment; it is therefore related in the broad sense of extending FNO, yet distinct in mechanism and problem setting (Zhao et al., 2023).

A common misconception is to treat MFNO as a generic label for any FNO variant that improves performance on difficult inputs. The published formulation is narrower. MFNO refers specifically to the mirror-padded construction introduced for non-Markovian stochastic processes, with theory built on Wong--Zakai regularization and Sobolev-space universality. In that sense, its contribution lies at the intersection of operator learning, stochastic analysis, and boundary-compatible spectral modeling rather than in a generic enhancement of Fourier layers.

7. Significance and scope of the MFNO formulation

The MFNO formulation combines three elements that are rarely assembled simultaneously in stochastic learning: a global operator architecture, a mathematically compatible treatment of non-periodic inputs, and rigorous approximation theory for non-Markovian dynamics. Its theoretical contribution is not a direct approximation theorem for arbitrary rough stochastic maps; instead, it proceeds by learning the continuous surrogate induced by Wong--Zakai regularization and then quantifying the gap to the true stochastic dynamics (Lee et al., 23 Jul 2025).

In practical terms, MFNO learns stochastic dynamics as operators on paths rather than as pointwise sequences. This is particularly appropriate for systems in which the future depends on the entire past. The reported experiments indicate that this formulation can support accurate prediction across discretizations and fast generation of sample paths after training.

The present formulation is therefore best understood as a specialized adaptation of FNO to stochastic path space. Its central innovation is not a new Fourier kernel parameterization, nor an attention or multiscale hierarchy, but the insertion of mirror padding and truncation so that non-periodic stochastic inputs can be handled within a periodic Fourier-operator framework without violating the Sobolev regularity assumptions on which the approximation theory depends.