Mesh-Informed Neural Operator : A Transformer Generative Approach
Authors: Yaozhong Shi
Zachary E Ross
Domniki Asimaki
Kamyar Azizzadenesheli
Venue: Accepted by TMLR ()

Abstract: Generative models in function spaces, situated at the intersection of generative modeling and operator learning, are attracting increasing attention due to their immense potential in diverse scientific and engineering applications. While functional generative models are theoretically domain- and discretization-agnostic, current implementations heavily rely on the Fourier Neural Operator (FNO), limiting their applicability to regular grids and rectangular domains. To overcome these critical limitations, we introduce the Mesh-Informed Neural Operator (MINO). By leveraging graph neural operators and cross-attention mechanisms, MINO offers a principled, domain- and discretization-agnostic backbone for generative modeling in function spaces. This advancement significantly expands the scope of such models to more diverse applications in generative, inverse, and regression tasks. Furthermore, MINO provides a unified perspective on integrating neural operators with general advanced deep learning architectures. Finally, we introduce a suite of standardized evaluation metrics that enable objective comparison of functional generative models, addressing another critical gap in the field.

Published in Transactions on Machine Learning Research (10/2025)
Mesh-Informed Neural Operator : A Transformer Generative Approach
Yaozhong Shi
yshi5@caltech. edu
California Institute of Technology
Zachary E. Ross
2ross@caltech..om edu
California Institute of Technology
Domniki Asimaki
dommikk@@alttcch.he edu
California Institute of Technology
Kamyar Azizzadenesheli
kaazizzad@ggaiil.com com
NVIDIA Corporation
Reviewed on OpenReview: https: // opwnreview......... net/ forum? id- K8qAuRfuOG
Abstract
Generative models in function spaces, situated at the intersection of generative modeling and operator learning, are attracting increasing attention due to their immense potential in diverse scientific and engineering applications. While functional generative models are theoretically domain- and diseretizatioonaanosiiic current implementations heavily rely the Fourier Neural Operator (PNO), limiting their applicability to regular grids and rectangular domains. To overcome these critical limitations, we introduce the Mesh-Informed Neural Operator (MINO). By leveraging graph neural operators and cross-attention mechanisms, MINO offers a principled, domain- and discretizattion-agussii backbone for generative modeling function spaces. This advancement significantly expands the scope of such models to more diverse applications in generative, inverse, and regression tasks. Furthermore, MINO provides unified perspective integrating neural operators with general advanced deep learning architectures. Finally, we introduce suile of standardized evaluation metries that enable objective comparison of functional generative models, addressing another critical gap in the field.
1 Introduction
Generative models are powerful tools for fields dealing with complex data distributions, with recent advanies in diffusion and flow matching models demonstrating impressive capabilities for synthesizing high-fidelit
images (Song et al. 2021 Ho et al. 2020 Lipman et al. 2023), audio (Liu et al. 2023 Huang et al. 2023), and
video (Jin et al. 2025). These models excel at learning highly complicated probability distributions in finite- as seismology, biomechanics, astrophysics and atmospheric sciences primarily deal with data that inherently reside in infnite-dimeesionall function spaces. Moreover, in many cases, the data collected in these fields are functions sampled on heterogeneous networks of sensors, Earth's surfacc). Functional generative models are especially important for these fields because they routinely deal with several broad (sometimes overlapping) challenges: (i) uncertainly quantification in physical units, (ii) the inherent non-uniqueness present in, e.g. solutions to inverse problems, and (iii) the need to model stochastic latent fields that are effectively unobservable, which are better handled probabilistically. Together, these factors necessitate a paradigm shift towards generative models that operate directly function spaces.
dimensional spaces. However, numerous fields in science and engineering
manifolds (e.g., global seismic networks weather stations

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Recent studies have generalized various generative paradigms to function spaces Rahman al 2022
Seidman et al. 2023 by leveraging neural operators |Aizzadeneshelli et al. 2024 Kovachki et al. 2023 Despite these advancements, two critical bottlenecks hinder the broader adoption and rigorous evaluation of functional generative models. First, current implementations are predominantly based on FNO (Li et al. 2021 as their backbone. However, FNO implementations are restricted to regular grids on rectangular domains, thereby preventing the realization of many key theoretical benefits. Second, prior studies rely on dataset-specific metrics, making it difficult. to compare functional generative models across datasets and to draw robust conclusions about generative maaitv
Shi et al. 2024a by Kerrigan et al. 2023a Lim et al.
2025
Cross-Attention Block
Geometry Encoder
Latent Processor (optional)
Inout function Usio Narl
Latent
Cross Attention
GNO
Q KV
Q KV
Phentnta: Reatlnward
Positions [Pamn. Maa]
Pattr Massel
emeatejp T& other canditiun
Emboc
Soae
[Caan]
Cross Attention
Output function
KV
Po.Po
Cross-Attention Dacoder
Overview of the MINO architecture. The geometry encoder uses a GNO as domain-agnostic tokenizer, followed by several cross-attention blocks and an optional latent processor. The decoder then employs a distinct cross-attention mechanism to map the latent reprosentation back to the target locations.
Figure 1:
This study addresses the aforementioned limitations of functional generative models. Specifically, our contributions are summarized as follows:
We introduce MINO, a domain-agnostic functional generative backbone. Our comprehensive ex- periments show that MINO achieves state-of-the-ar (SOTA) performance on a diverse suite of benchmarks with regular and irregular grids.
We demonstrate through analysis and experiments that Sliced Wasserstein Distance (SWD) and Maximum Mean Discrepancy (MMD) are efficient, robust, dataset-independent metrics for evaluating the performance of functional generative models on regular and irregular grids.
We propose a novel framework that integrates operator learning with modern deep learning practice. By using a graph neural operator as a fomain-agnostic tokenizer and embedding tailored cross- attention modules in both the encoder and decoder, our design avoids information bottlenecks present in prior
preceer neural operators, allowing any powerful, finie-ddmensional network (c.g., U-Net or Transformer) to serve the latent procossor.
2 Preliminaries
Neural operators. Neural operators learn mappings between function spaces, extending deep learning beyond the fixed-dimensional vectors handled by classical networks. This function space perspective is especially important for scientific computing that is governed by partial differential equations (PDEs) (Kovachki et al. 2023 Li et al.) 2021). A core theoretical property neural operators is discretization convergence (agnosticism): as the mesh of input function refined, the prediction approaches the unique continuous solution (Kovachki et al. 2023). The Fourier Neural Operator (FNO) (Li et al., 2021) achieves this property via spectral-domain convolutions, achieving quasi-linear complexity on regular grids. Its global mixing excels
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capturing long-range dependencies but may miss fine-scale detail (required for generative tasks) unless many Fourier modes are retained (Luu-Schiafini et al. 2024).
For handling functions irregular grids, Graph Neural Operator (GNO) Gilmer et al. 2017 Li al. 2020) remains a powerful candidate. GNO shares the message-passingg functionality of Graph Neural Network (GNN), but differs in that the search radius for neighbors is defined in physical coordinates and made consistent across resolutions to guarantee discretization convergence. This results in GNO having a local receptive field, however it is computationally inefficient for certain tasks (Li et al. 2023; Liu-Schiaffin et al. 2024) and often struggles to learn the high-frequency information required for many generative models. This difficulty arises from the standard message- muessage-psssng mechanisms, which tend to act as low-pass filters (Li et al. 2018 Giovanni et al.) 2023 More recent work has focused on improved scalability and performance aspects of Neural Operators on irregular grids and non-rectangular physical domains, e.g., Geometry-Informet Neural Operator (GINO), Universal Physical Transformer (UPT), Latent Neural Operator (LNO), and
Transolver (Li et al. 2023 Alkin et al., 2024; Wang & Wang 2024 Wu et al. 2024). However, it remains unclear whether these architectures will also be effective for functional generative modeling on irregular grids (scc Appendix
Functional generative models. Classical generative models learn a mapping from a simple base distribution (e.g,, multi-variate Gaussian) to a target finite dimensional data distribution. Alternatively, the functional perspective frames this problem in dnfnite-dimenssonall spaces. A sample function fi D - Ra-,, with D denoting the domain of the function, is treated as a sample from target probability measure /a over function space. The discrete samples are formed by evaluating fi on set of grid points of locations. Functional generative models often aim to learn a mapping between function spaces by training a neural operator to transform a function fo sampled from simple base measure μo (typically a Gaussian random field mcasure (0, C)) into a function fi that follows the target mcasure /41 The distribution of the discrete data is thus a finite-dimensional marginal of the true data measure P41
Adopting this perspective, functional generative models offer several key benefits not generally available in their traditional fiiteediimensional counterparts:
Flerible domain geometries: The function domain D is not required to be rectangular; it could be an irregular domain or even a manifold. Such flexibility ideal for many real-world scenarios in science and engineering, such as modeling rainfall over a city with an irregular boundary, or modeling weather patterns on the Earth's surface.
Discrelization agnoslicism (mesh (nvariance): By learning a probability measure over functions, func- tional generative models can be trained on samples with varying discretizations and can subsequently generate new function samples at any desired discretization in a zero-shot manner.
Inference-time control and guidance in function space: Functional diffusion/flol /flow models enable the incorporation of inference-tine scaling rules and precise guidance signals directly within function spaces. For instance, this facilitates the development of probabilistic PDE solvers that can rigorously enforce hard boundary conditions or other external controls during generation (Cheng et al. 2024
Yao et al. 2025).
Universal functional regression via stochastic process learning: By leveraging invertible trajectories derived from neural operators, it is possible to perform functional regression with learned (non-
Gaussian) stochastic process priors (Shi et al.) 2024a 2025 These learned distributions are capable of providing exact prior and posterior density estimation for a general stochastic process.
Flow matching in Hilbert space. Recent work has established a rigorous mathematical framework for extending flow matching to Hilbert spaces (Kerrigan et al. 2023b), Functional Flow Matching (FFM), where a velocity field learned to transport a base Gaussian mcasure to a target measure. This was further built upon by Operator Flow Matching (OFM) (Shi et al., 2025). which extended the paradigm to stochastic processes and incorporated dynamic optimal-transport path.
The core idea of FFM/OFM is to learn continuous path of functions, fi for E [0, 1), that transforms samples So from base Gaussian measure Hi into samples fi from a target data measure fa- This transformation is
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governed by Ordinary Differential Equation (ODE) whose velocity field, ve, is parameterized by neural operator with weights 0:
dft (fc.t)
(1)
dt
For training, OFM draws pairs of functions (fo. f1) from a mini-batch optimal coupling I T(H, /2), which is achieved by minimizing the 2- 2-Wasserstein distance between ofl and f41. The base measure, μo, is typically a Gaussian measure (O0 C)2 where C' is a trace- class covariance operator. The framework then defines the path ft as a linear interpolation between fo and fi, i.c., ft = (1 fo + tf-- Se induces a time-dependent probability measure Pt, such that at = 0, ur - po and = 1, Me pt- This specific choice yields a simple ground truth expression for the velocity, e = Si fo- The training objective is then minimize the Mean Squared Error (MSE) between the parameterized velocity field vo(fi, t) and this target U,. To generate new sample, one draws fo dAo and solves the learned ODE numerically. While functional diffusion models offer a related framework, they lypically rely on Stochastic Differential Equations (SDEs), which introduce an additional perturbation term with longer generation time and inferior performance compared to ODE-based flow matching (Lim et al.) 2025 Kerrigan et al. 2023a). fundamental property of both paradigms is domain aligmment: for a given trajectory, all evolving functions fo must be defined on the same fixed spatial domain D. (14 geometrically complex; it is not restricted to rectangular shape and can be a domain with an irregular boundary or even a manifold. However, this theoretical ficxibility has been underutilized in practice, as existing implementations predominantly rely on FNO, which restricts them to domains discretized by regular grids.
(D;RRam))) Crucially, the theory allows this domain D to be
Grid-Based Architectures. The modern U-Net architecture used in diffusion models (Rombach el al., 2022 has become de-facto standard for generative tasks on grid-like data. These advanced U-Nets are distinct from the variant used in some carlier neural operators (Rahman et al., 2023). the U-Net can effectively capture multi-scale information through their hierarchical structure, skip connections, and integration of attention and residual blocks. Nevertheless, they remain constrained to regular grids due to their reliance on standard convolutional neural network kernels. Another prominent line of work involves Transformer- based architectures. Standard Vision Transformers (ViTs) and Diffusion Transformers (DiTs) (Dosovitskiy et al. 2021 Peebles & Xie 2023), first. "patchily" the input into non-overlapping blocks and then apply full self-attention, incurring O ((tpht))) time complexity. For images, NNpatch < H W (Height and Width), yet the patchify operation creates discontinuities at block borders that can lead to checkerboard artifacts (Fang et al. 2023 Moreover, both VIT and DiT presuppose fixed rectangular grid so that patchification is well defined.
Table 1: Comparison of MINO with other architectures. "local - global receptive field gathers global context through stacked local operations. "local global' receptive field combines local and global information directly. A detailed discussion is provided in Appendix C
Architecture
Efficient
Receptive field
Irregular grid Discretization convergent
Frequency learnt
GNN
local
low
GNO
local
low
FNO
global
X
low + high
U-Net
local + global
X
X
low + high
VIT
global
X
X
low + high
GINO
local + global
low
UPT
local + global
low
MINO [Ours]
local + global
/ low + high
3 Methods
Our proposed architecture, MINO, provides an effective solution for functional generative models on non- rectangular domains with irregular grids. It integrates powerful deep learning backbones with neural operators in manner that is tailored for improving functional generative performance. Specifically, this is achieved
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(a)
(GP nolse)
T=
(output)
Inference Time
N-S
Weather
MeshGP
Cylinder
(b)
Zero-shot generation different spatial scales
Fine
Medium
Coarse
Figure 2: Inference and zero-shot Generation with MINO. (a) MINO gradually transforms a GP sample to the data sample under flow matching paradigm. (b) Zero-shot generation at varying spatial scales by directly transforming finer GP samples finer data samples.
combining GNO with cross-attention mechanisms (see Table Dp. As illustrated in Figure D the GNO in the encoder can be viewed domain- and discrelizatiion-agnsiiii alternative to the "patchify" operation. By injecting the input function representation into both the encoder and the cross-attention decoder, framework benefits from both local and global receptive fields.
In the following, we outline the MINO architecture. We specifically focus on its use in the flow matching paradigm to learn velocity fields, as illustrated in Fig. 2 Although our primary application is generative modeling, the MINO architecture general-purpose and can be used for other operator learning tasks, such as solving PDEs, with minor adjustments.
Problem Formulation. Within the flow matching paradigm, MINO takes three primary inputs: a noisy function f, the of its observation positions fpos. and the corresponding time step t, along with any optional conditioning variables. Specifically, the input function is represented by its values at Ni, discrete
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locations, fpos = {pr) These locations are discretization of continuous domain D C R Paim, Note that Ni can be different per sample. The codomain (channel) of f has dimension Sdim; resulting in an input tensor of function values with shape [fam. Nm.. The output is another function (velocity field ve) at these same locations. In the context of flow matching or diffusion, the output dimension matches the input, so the larget putput also has a shape of [9dmm Nul, where Idim = fdim-
The MINO framework is inspired by the encoder- prouder-processo-deeddeer structure of prior architectures like GINO and UPT. Our model consists of three main components: (i) geometry encoder, (fi) an (optional) latent-space processor, and (iii) cross-attention decoder. We introduce several key modifications to this design that significantly improve performance on functional generative tasks, detailed in the following subsections.
Geometry Encoder. First, we apply sinusoidal embeddings (Vaswani et al. 2017 Peebles Xie 2023 to
the time step and the observation positions fpos; yiclding temb and Pemb respectively, as shown in Eq. 2 The input function f is then concatenated with the embedded positions Pemb along the codomain before being passed to a GNO layer; this maps the function from its potentially irregular discretization { Ni, to Nande The GNO mapping is basod
latent representation on a predefined regular grid of query points {p² (pquery), neighbor-searchin within fixed radius r, where Nuode < Nin- Following this GNO layer, linear layer projects its output f. into hy of shape [Nodl-
1j=1 13-1
This initial latent representation he is then processed by M1 blocks of Multi- Head Cross-Attention (MHCA). Importantly, the key-value (KV) pair for all attention blocks is fixed as he while the query (Q) is the output of the previous MHCA layer 1 and the time embedding temb is used for conditioning. We found this configuration yields better performance than the standard self-attention used in DiT, with comparisons provided in Appendix Thus, for this entire component of the architecture, we have the following,
Pemb Emb(fpos), temb Emb(t)
(2)
E GNO(concat Pemb)), hy Lincar(fe))
(3)
hy = MHCA(Q h KV hp C tamb),
j 1= Mi.
(4)
The output of the geometry encoder has a fixed size of [Tdimm: Nuodel, which is independent of the manner in which the input function f, is discretized.
Latent-Space Processor. Given that the latent representation hM, has a fixed size, any suitable neural network (NN) can act as the processor. In our implementation, we use a Diffusion U-Net, known for being a strong backbone in generative modeling. It is important to note that this processor is optional; MINO remains well-defined neural operator without it, although found that its inclusion generally improves performance. The processor's input is conditioned temb, whereas its output, hP, maintains the same shape as its input,
P = NN(h temb)
55)
Cross- Cross-Attention Decoder. The decoder employs a different cross-attention mechanism from that of the encoder to enable mapping the latent representation back into the output velocity field at the original observation locations. Specifically, fę is first processed by MLP and then concatenated with Pemb to form fP. In a single cross-attention block, rP serves the query (Q), while the key-value (KV) pair is taken from the outpul of the optional processor hP (or hy, if the processor is omitted). The time embedding temb provides conditioning. Finally, a LayerNorm and linear transformation are applied to the output of this attention block to produce the estimated velocity field vt.
PD = MMLP(concat Pemb))
(6)
hp = MHCA(Q = r.
KV 1 -V= C temb)
(7)
Vt = Linear (Lyear((ayeNNorm(hh)
(8)
A detailed comparison of the MINO framework to prior neural operator architectures is provided in Appendix with ablation study of our model's components in Appendix F

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4 Experiments
In this section we empirically evaluate MINO and other baselines on a suite of functional generative benchmarks under the OFM (Shi et al., 2025) paradigm due to its concise formulation and SOTA performance among functional generative paradigms. For each task we paramcterize the velocity field vo(ft,t) with different. neural-operator backbones study how the choice of architecture alfects performance. The benchmarks cover both regular and irregular grids, providing comprehensive assessment. To the best of our knowledge, this is the first systematic comparison of modern neural-operator architectures on Ainetion-geneattin problems. We evaluate two variants of our proposed architecture: (i) MINO-U: a Diffusion U-Net as the latent space processor, and (ii) MINO-T: a pure Transformer-based variant without the latent space processor, where the encoder receives additional cross-attention blocks.
Table 2: Summary of experimental beuchmarks, covering diverse domains with both regular and irregular grids
Geometry
Datasets
Domain
Co-domain
Mesh
Training Samples
Test Samples
Navier Stokes
D C g²
1
4,096
30,000
5,000
Regular Grid
Shallow Water
D C R²
1
4,096
30,000
5,000
Darcy Flow
D C R²
1
4,096
8,000
2,000
Cylinder Flow
D C R²
3
1,699
30,000
5,000
Irregular Grid
MeshGP
D C R²
1
3,727
30,000
5,000
Global Climate
g2 R3
1
4,140
9,676
2,420
Datasets. We curated a benchmark of six challenging functional datasets to evaluate performance across
different domain types: (i) Rectangular Domains and Regular Grids: We use Navier-Stokes (Li et al. 2021]
Shallow Water equation (Takamoto et al. 2022). and Darey Flow (Takamoto et al., 2022) datasets. (ii) Irregular Domains and Grids: We use the Cylinder Flow dataset (Han et al. 2022), a synthetic Mesh-GU 2022). and rcal-world global climate dataset (Dupont et al.
datasct on irrogular meshes (Zhao et al.)
2022). These datasets are sunmarized in Table Detailed descriptions of the dalasets and their associated preparation are provided in Appendix
Baselines. To ensure thorough comparison, we selected several state-of-the-art neural operators tailored to different domains: (i) For irregular grids, we benchmark against leading architectures including the Universal Physical Transformer (UPT) (Alkin et al. 2024), Latent Neural Operator (LNO) (Wang & Wang 2024). Transolver (Wu et al., 2024h, and Geometry- Informmmd Neural Operator (GLNO) (Li et al. 2023). (ii) For regular grids, we further include the Fourier Neural Operator (FNO) as a strong, established bascline. To ensure fair assessment, the parameter counts (or computational budgets) of all baseline models were carefully matched to our variants. It is important to note that, with the exception of FNO, all evaluated models (including our MINO variants) are designed to be domain-agnostic and process regular and irregular grids identically. A detailed description of the the baseline implementation is provided in Appendix A
Metrics. A significant current challenge in the field of functional gemerative models is the lack of standardized evaluation protocols. Previous studies often rely on dataset-specific metrics, which prohibits fair and direct comparison between models (Rahman et al.) 2022 Kerrigan et al.] 2023a Shi et al.] (0224a] Lim al. 2025). To address this gap, our work validates two robust and general metrics suitable for estimating the distance between the learned probability measure (from which generated samples X drawn) and the target. probability measure μ (from which test samples (dataset) Y are drawn).
The first metric is the Sliced Wasserstein Distance (SWD) (Bonncel et al. 2015 Han 2023 SWD provides a compulationally efficient yet powerful alternative to the exact Wasserstein distance, which is often intractable in high dimensions. It works by projecting nigh-dimensional distributions onto random 1D lines and then averaging the simpler 1D Wasserstein distances. This approach converges to the true Wasserstein distance as the number of projections increases and provides a robust way to compare the geometric structure of two probability measures. For two measures μ and v in Hilbert space, SWD is defined as (Definition 2.5 of Han
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2023
1
/ W7(0 8 (ss)s()))))))))
VWD =
(9)
Ts((S) |011=1
Where base measure Ts is strietly positive Borel measure defined on a sphere S, W, is the Wasserstein-pp distance, and 04 2 denote the pushforward measure of μ. v under the projection 0 separately. Further details are provided in Appendix B
The second metric is the Maximum Mean Discrepancy (MMD) (rreton et al. 2012) Let X'= = (2; Baam
and choose a bounded, characteristic kernel : X' x X - R with reproducing kernel Hilbert space (RKHS)
(Hgs (x). Denote the mean embeddings my E.A((X, )), mn E[](Y,))]. Then MMD defined as
MMD Y) my mm H
(10)
Because k is characteristic, MMD, if and only if 1 v. In Appendix B we show how to calculate SWD and MMD given discretized function samples, and empirically verify that these metrics are valid, robust, and sample-efficient for functional generative models.
Results. A visualization of the performance of MINO is provided in Figure 3 and as shown in Table regular grid benchmarks, our MINO variants achieve SOTA performance: MINO-U achieves best performance on the Navicr-Stokes and Shallow Water datascts, while MINO-T excels on the Darcy Flow benchmark. In contrast, several baseline neural operators (GINO, LNO, UPT) perform poorly. Further analysis in Appendix C D suggests this because these models primarily learn low- www-frquency information and fail to capture the high-frequency components essential for these generative tasks. FNO and Transolver achieve comparable, albeit inferior, results to our models. Notably, beyond superior accuracy, our MINO variants are also significantly more efficient. than Transolver. As detailed in Table 5 they require substantially less computation to train, are more GPU memory-efficient, and achieve up to 3.0x speedup during inference, demonstrating the efficiency of the MINO architecture.
For tasks on irregular grids, our models continue to demonstrate top performance. MINO-U achieves the best results on the Cylinder Flow benchmark, while MINO-T is the top performer on the Mesh-GP and Global Climate benchmarks. Then, We perform an ablation study, which confirm the effectiveness the each component of our architecture, detailed in Appendix Last, provide additional experiments in Appendix F and to further demonstrate the superior performance of MINO.
(a)
Ground
(b)
Ground Truth
MINO Super Resolution
MINO
Figure 3: Visualization of generation and zero-shot super-resolution by MINO-U. (a) Navier-Stokes samples generated on the original mesh (4,096 nodes) and a finer mesh (25,600 nodes). (b) Global Climate sample generated on the original mesh (4,140 nodes) and finer mesh (16,560 nodes).
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Table 3: Generation performance on benchmarks with irregular grids. Best performance bold
Dataset
Cylinder Flow
Mesh- GP
Global Climate
Model | Metric +
SWD
MMD
SWD
MMD
SWD
MMD
GINO
4.6-10-1
3.1 10-1
1.1-10 0
3.6 10-1
6.9-10-1
5.1 10-1
UPT
7.3
1.4-10-1
4.2-10-1
2.8 10
7.4-10-1
5.4 10
Transolver
2.5 2.5-10-
204-10-2
9.1-10-2
4.6.10-2
2.8-10-2
2.9 10
LNO
5.3-10-1
30--10-1
3.5-10-1
2.4-10-1
6.7-10-1
4.9 10-1
MINO-T (ours)
2.9-10 10
2.6-10
4.1
3.0- 10
2.1-
1.4-10
MINO-U (ours)
2.3 10
2.1
2.1-10-2
7.2
10-2
4.1.10
10-2
2.8-10-2
2.6- 10-2
Table 4: Generation performance on benchmarks with regular grids. Evaluation metrics include the Sliced Wasserstein Distance (SWD), Maximum Mean Discrepancy (MMD), and Mean Squared Error (MSE) for spectra, autocovariance, and point-wise density. Best performance is indicated in bold.
Datasets
Model Metric
SWD
MMD
Spectra-MSE Autocovariance-Mee MSE Density-MSE
GINO
4.6
3.7.10"
1.9 102
4.6 10
1.9 10
UPT
6.0
4.7-10-1
2.5 10
1.5 10-10-2
4.0- 10
Transolver
50-2
5.1-10-2
1.1-
6.7-
2.0-10-1 10
Navier-Stokes
LNO
5.2-
3.8-10-1
2.0
1.1-10-1
5.0- 10-4
FNO
3.2
2.6-10-2
5.1-10"
2,2-10
6.9 10-5
MINO-T (Ours)
4.0-10-2
3.6-10-2
1.9-10
6.8-10-5
1.6- 10-5
MINO-U (Ours)
2.8-10-2
19-10-2
1.3 1.3-10' 10"
9,2 9,2-10-1
14 14-10-5
GINO
103
5.1-10-1
1.5
1.8110-1
10-1
UPT
806-10-1
5.9-10-1
3.1
4.0 40-3
1.0
Transolver
1.7
18-10-2
2.0-10-2
1.2-10
10-1004
Shallow Water
LNO
80-
6.5-10-1
2.1-102
1.4
1.0-10
FNO
10.10-2
9.4-10-3
7.9-10-3
2.8-10-6
1.0-10-1
MINO-T (Ours)
9.8-10
8.7-10
16-10-2
5,2 10-6
2.0-10 10-10%
MINO-U (Ours)
4.0-10
1.1-10
4.3
5.1 10
-
GINO
4.0-10-1
3.0-10-1
5.5-10
10-2
1.8- 10-3
UPT
8.7-10-1
4.6-10-1
6.0
4.
7.5-10-3
Trransovvrr
2.2.10-1
8.9-10-2
3.4-10
3.0 10-4
Darcy Flow
LNO
5.5
30-
2.7
2.4 20-
1.0 10
FNO
1.1
30--
1,2-103
8.4 10-2
8.9 10
MINO-T (Ours)
8.9 10
30-
9.1 10a
4.4 14.10-2
9.4 10
MINO-U (Ours)
9.5 10
4.8-1
1.0-10
7.7 17-12-2
3.8 10
Table 5: Computational efficiency and performance comparison between MINO variants and Transolver during training and inference on Navier-Stokes benchmark.
Training Phase->
Params
GPU memory (per sample)
Training time (per epoch)
Speedup
Transolver
15.0 M
0.913 GB
734
1x
MINO-T (ours)
21.5 M
0.429 GB
277
2.6x
MINO-U (ours)
19.2 M
0.419 GB
279
2.6x
Inference Phase
SWD
MMD
Generation time (per sample)
Speedup
Transolver
0.053
0.051
1.49
1
MINO-T (ours)
0.040
0.036
0.51 s
2.9x
MINO-U (ours)
0.028
0.019
0.49
3.0x
5
Conclusions
In this paper, we introduce the Mesh-Informed Neural Operator (MINO), a novel backbone for functional generative models that leverages graph neural operators and cross-attention mechanisms. MINO operates

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Published in Transactions on Machine Learning Research (10/2025)
directly arbitrary meshes, addressing the reliance grid-dependent architectures like FNO and thereby enabling high-fidelity generation on complex and irregular domains. To complement this architectural advance, we validate the Sliced Wasserstein Distance (SWD) and Maximum Mean Discrepancy (MMD) as general-purpose metrics for fair and standardized model comparison. Our comprehensive experiments confirm the success of this approach: MINO variants achieve state-of-the-arl performance across a diverse suite of benchmarks while being substantially more computationally efficient than strong competitors.
Additionally, the MINO framework helps bridge the gap between math-driven operator learning and modern deep learning practice, unlocking the potential to apply high-fidelity generative modeling to broader range of complex scientific problems. We believe contributions will foster more rigorous and rapid advancement in the field of functional generative modeling, Python code is available at MINO
com/yzshi5/
Acknowledgments
The authors would like to thank the TMLR reviewers and the action editor for their constructive feedback, and in particular reviewer fkny for the valuable suggestion regarding Table g This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Science Foundations for Energy Earthshot under Award Number DE-SC0024705, ZER is supported by a fellowship from the David and Lucile Packard Foundation.
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A Experimental setup
A.1 Datasets description
We curated benchmark of six challenging functional datasets to evaluate performance across different domain types, as summarized in Table
Navier-Stokes. This dataset contains solutions to the 2D Navier-Stokes equations on a torus at resolution of 64 x 64. Following the pre-processing of previous work (Kerrigan et al., 2023b; Shi et al. 2025 we use
30,000 samples for training and 5,000 for testing, drawn from the original dataset introduced (Li et al., 2021).
Shallow Water. This dataset contains solutions to the shallow-watee equations for 2D radial dam-break scenario on square domain, from PDEBench (Takamoto et al. 2022 Each of the 1,000 simulations has 1,000 time steps at 128 x 128 resolution; we downsample spatially to 64 x 64 for efficiency and treat each time step as an independent snapshot. We randomly select 30,000 snapshots for training and 5,000 for testing
Darcy Flow. This dataset contains steady-state solutions of 2D Darcy Flow over the unit square, obtained directly from the PDEBench benchmark (Takamoto et al. (022). We downsample the original 128 128 resolution to 64 x 64. The dataset. contains 10,000 samples and we split it into 8,000 samples for training and 2,000 for testing.
Cylinder Flow. We use the Cylinder Flow dataset of Han et. al. 2022 which describes flow past cylinder on a fixed mesh of 1,699 nodes. Each sample is a 3-channel function (x-velocity, y-velocity, pressure). From 101 simulations x 400 time steps, we ignore temporal order and treat each time step as an independent sample, randomly selecting 30,000 training and 5,000 testing samples.
Mesh GP. This is a synthetic dataset generated on a fixed irregular mesh of 3,727 nodes provided by (Zhao et al. 2022). We generate function samples from a Gaussian Process (GP) with Matérn kernel (length scale 0.4, smoothness factor- 1.5) given the domain, creating training set of 30,000 samples and test set of 5,000 samples.
Global Climate. We use the real-world global climate dataset from (Dupont et al. 2022). which contains global Lemperature measurements over the last. 40 years. Each data sample is function defined on a grid of 46 x 90 evenly spaced latitudes and longitudes. Following the previous pipeline (Dupont et al. 2022). we convert the latitudo-longitude pairs to Euclidean coordinates (R²) before passing them to the models. The dataset contains 9,676 training samples and 2,420 Lest samples.
A.2 Baselines description
For all baseline models, we adopt their official implementations to ensure reproducibility; the corresponding code repositories are linked in the referenced papers. The key initialization hyperparameters for the baselines GINO [Li et al. 2023), UPT (Alkin et al., 2024), Transolver (Wu et al.) 2024). LNO (Wang & Wang 20244), and FNO (Li al. 2021) are detailed in Table 6 Note that slightly dilferent configuration was used for
the Global Climate experiment. For a complete understanding of all arguments, refer the reader to the official repository for each respective model (github link provided in all referred papers)
A.3 Hyperparameters for MINO
For the decoder, the single cross-altention block followed by a multi-head self-attention (MHSA) block. Although this MHSA block is optional, we found that its inclusion slightly improves performance for generation task. To maintain the overall lincar time complexity of MINO, the MHSA module can be replaced by any self-attention variant with linear complexity (like transolver layer).
MINO-T. The GNO maps input functions to latent representation on 16 x 16 grid of query points, defined over the [0, 1|2 domain. We set the GNO search radius to 0.07, the latent. dimension Lain to 256, and the number attention heads to 4. The encoder consists of M1 = 5 cross-attention blocks. For the Global Climate dataset, which is defined on a spherical manifold S? we adjust the latent query positions to a x 16 spherical grid and increase the GNO radius to 0.2 to account for the different coordinate system; other hyperparameters remain unchanged. The total parameter count for MINO-T is 21.5 M.
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Table 6: Hyperparameter settings for all baseline models.
Hyperparameter
GINO
UPT
Transolver
LNO
FNO
Parameters
19.7 M
19.6 M
15.0 M
22.2 M
20.6 M
GNN radius
0.07
GNO radius
0.06
Width/ /ii.
128
192
512
512
128
Blocks LLayes
4
12
10
8 X
4
Attn. Heads
3
6
Latent Grid
32 x
Latent Tokens
256
256
Fourier Modes
16
24
FNO Channels
180
128
Slice Number
24
MINO-U. The GNO maps input functions a latent representation on a 16 16 grid of query points, defined over the [0, 1 domain. We set the GNO search radius 0.07, the latent dimension Laim to 256, and the number of attention heads to 4. The encoder consists of M1 cross-attention blocks. For its latent-space processor, adopt a Diffusion U-Net architecture from the torchefm library (Tong et al. 2024 which operates on the [16, 16] latent tensor with 64 channels, 1 residual block, and 4 attention heads for the processor. Same sctting of MINO-T on the Global Climate datasct. The total paramcter count for MINO-U
is 19.2 M.
A.4 Details for training and inference
Reference Mcasures. We choose as Gaussian mcasure characterized by a Gaussian Process (GP) with a Matérn kernel. Unless otherwise specified, the function domain is [0, 1]2 (or subset of it), and we use a kernel length scale of 0.01 and a smoothness parameter of 0.5. For the Global Climate dataset, the domain a spherical manifold represented as a subset of 1],, and the kernel length scale adjusted 0.05. Furthermore, for the GP on the sphere, we explored two distance metrics for the kernel. The first uses the chordal distance (the Fuclidean distance in the TR3 embedding space), while the second employs the geodesic distance on 82, for which we leverage the Geometri ceennel library (Mostowsky et al. 2024h Although both approaches yield a valid GP, the results presented in this paper are based on the chordal distance implementation. We selected this method for two primary reasons: its superior computational speed and to ensure consistency with other benchmarks. To facilitate further exploration, we will release the codc repository with both implemenntations. The variance is fixed at for all GPs.
Training Details. We train all models for 300 epochs using the AdamW optimizer with an initial learning rate of 1c-4. We employ a step learning rate scheduler that decays the learning rate by gamma of 0.8 every 25 epochs. The default batch size is 96. However, Transolver consumes significantly more GPU memory than other models. To accommodate it on single NVIDIA RTX A6000 Ada GPU (48 GB memory), we reduced its batch size to 48 and, to maintain a comparable training iteration (duration), limited its training to 200 epochs. Despite these adjustments, Transolver still required approximately 1.76x more total GPU-hours than MINO-T and MINO-U, respectively. The only exception was the Global Climate experiment, where we explicitly matched the total GPU computation time of MINO variants to that of Transolver by extending their training epochs. All experiments reported in Table | and Table 3 were conducted three times, and we report the best performance (among the three) achieved for each model. To further strengthen our comparative analysis, Appendix G details an experiment with matched settings (parameter counts, batch size, epochs) that reveals MINO's significant speedup and superior performance over Transolver.
Inference Details. To evaluate the models, we generate the same number of samples as contained in the test set for cach dataset shown Table 2 All samples are generated by solving the learned ODE numerically using the dopri5 solver from the torchdiffed library (Chen et al. 2018), with an error tolerance set to 1c-5 for all experiments.
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B SWD and MMD as general metrics for functional generation tasks
Unlike image generation, where established metrics like the Fréchet Inception Distance (FID) can leverage well pretrained models to compare distributions in latent space, the field of functional generative models lacks such standardized evaluation protocols. Functional data covers a wide range of modalities, often without a common pretrained model, what's more, the learnt objective is probability measure not probability distribution, which requires the metrics should be consistent regardless of discretization of function sample. This makes fair and direct comparison between models challenging, as previous studies frequently rely on dataset-specific metrics. To address this gap, our work proposes and validates two general metrics that directly estimate the distance between learned probability mcasure, v, and target measure, H4, using samples drawn from each.
We are given two sets of i.i.d. function samples, X = ),
13=1 drawn from v and Y = 1 drawn from μ.. Each function is observed at the same Ni, locations. To compute the distance, each function sample is first flattened into a single vector representation.
Sliced Wasserstein Distance (SWD). In practice, we estimate the distance between two probability measure,4 and v, using the discretized function samples in the datasets X and Y. To make the metric shown need to derive the discretized version of it, which is achieved by choosing the base measure Y to be the Haar measure (uniform surface measure) on the sphere Rd-1 (d is the discreziation of functions and d = No. if co-domain is 1 in our case) and replace the integral and normalization with an expectation with respect to the probability measure defined by normalizing Ys- Finally, we can get a discretized version of Eq. g
in Eq. g computable,
SWD,(KgV> =
E (1
(11)
where 0 is random direction on the unit sphere gd-1 SWD converges to the exact p-Wasserstein distance for two probability measures as the number of projections goes to infinity. In experiments, we use the official implementation from the POT library (Flamary et al. 2021 We use the Sliced Wasscrstein-2 distance (p- 2) (Bonneel et al., 2015), the time complexity for SWD is O(L N Nin + L- N log N), where L is the number of projections.
Maximum Mean Discrepancy (MMD). For MMD (Gretton et al. 2012) the time complexity is O(N N(.) we use the Gaussian RBF kernel k(.,)). Given the sample sets X and Y, we compute the squared MMD using the standard unbiased U-statistic estimator:
M N(N - 1)
M) + N(N - 1) itj i4j
2 N
Σ
k(a
((
(12)
A practical consideration for the SWD is that it relies on a Monte Carlo approximation over random projections. This can introduce variance; calculating the SWD multiple times on the same two test datasets may yield slightly differont results. To ensure our evaluations are stable and meaningful, we must reduce this variance to a negligible level. We propose a simple procedure we term averaged-SWD: instead of a single SWD estimation, we perform Ryun independent estimations via Eq I1 each with a large number of projections,
256 in our case) and report the mean, we would expect averaged-SWD has very small variance introduced by the Monte Carlo approximation.
To validate this approach, created two distinct Gaussian measures, characterized by GP, GP1 (length scale=0.3, smoothnes==1.5) and GP2 (length scalc=0.01, smoothness=0.5), on a 64 x 64 regular grid. We drew 5,000 GP samples from each measure to create X, Y and then calculated the averaged-SWDD between them for varying Teun and repeated this entire process 20 times to report the mean and standard deviation of the avcragod-SWD, As shown in Table 7 when Drun = 10, the variance of the averaged-SWD is sufficiently small. We therefore adopt Nyun 10 for all SWD computations in our experiments. In contrast, MMD is deterministic for a given kernel and does not exhibit. this variance.
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Metric Consistency Across Discretizations. A crucial property of metric for functional data is its consistency with respect to the discretization of the function domain. The computed distance between two measures should be consistent regardless of the discretization for functions samples in two function datasets drawn from the two probability measure. That is, if we draw new function samples X, and Y from the same measures l and v but observe them on a different set of locations, we expect MMD(X+; Yf) MMD(X,Y) and SX, Y)) = SWD(X,Y).
To verify this property, we calculated SWD and MMD in three representative scenarios while varying the number of observation points by randomly sub-sampling the full discretization (Nvn). The scenarios are: (i). GP1 vs. GP2. (ii). GP1 vs. Navier-Stokes (test dataset) (iii) GP1 vs. Cylinder Flow (test datasct). For each case, we will unify the observed position (discretization) of function samples
As shown in Fig. 4 both MMD and SWD remain highly consistent across different sub-sampling ratios the obscrvation points. This analysis demonstrates that SWD and MMD are not only theoretically sound but also empirically robust and sample-efficien evaluation metries for functional generalive models.
Table 7: Variance analysis of the averaged-SWD metric. We report the mean and standard deviation of the averaged-SWDD, calculated over 20 trials, for different numbers of averaging runs (Nrun).
Aran
10
20
40
60
SWD (mean + std)
0.2482 + 0.0052 0.2485 + 0.0038
0.2494 + 0.0029
0.2497 + 0.0017
0.2498
0.0014
GP1 GP2
GP1 vs Navier Stokes
GP1 vs Cylinder Flow
1.0
1.0
swd
0.8
mmd
0.8
Metric value
0.6
0.6
0.6
0.4
0.4
swd
swd
0.2
0.2
mmd
mmd
0.0
0.0
0.0
1000
2000
3000
4000
1000
2000
3000
4000
500
1000
1500
Numberof discretization
Number of discretization
Number of discretization
Figure 4: Consistency of SWD and MMD with respect to the number of discretization points. The metrics remain stable the number of observation points is varied for three comparison scenarios: two synthetic GPs (left), GP vs. regular grid dataset (center), and GP vs. an irregular mesh dataset (right).
C Connection and Difference to Prior Works
C.1 Functional Generative Models vs. Surrogate PDE Solvers
While both functional generative models and surrogate PDE solvers utilize neural operators as their backbone architecture, the demands placed on the operator these two settings are fundamentally different. A surrogate PDE solver typically learns one-step mapping between function states, such as mapping initial conditions to final solution. In contrast, functional diffusion flow model must learn to parameterize continuous transformation between two probability measures, and also requires many steps evaluation during inference. This contrast is further highlighted by the signal frequencies involved: surrogate solvers often map between smooth, low-frequency functions (Li et al. 2023), ; while functional generative models must. transform high-fres figh-frequency noise into structured data samples. Consequently, designing a neural operator for gencrative tasks is significantly more demanding, as the learning task itself is more challenging. There are some dinite-dienniinnll generative models (Havrllla et al.) 2024 Wang et al. 2024) can also achieve zero-shot generation at arbitrary resolutions. However, these approaches do not explicitly learn probability measure
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over the function space and the zero-shot generation is similar to the linear or bicubic interpolation, which distinguishes them from the functional generative paradigm discussed here.
C.2
Architectural Comparison to Other Neural Operators
The MINO architecture builds upon insights from previous work, including GINO and UPT (Li et al. 2023
Alkin et al, 2024), but introduces several key modifications to address their limitations in the context of functional generative tasks.
Comparison with GINO. GINO adopts a GNO encoder and decoder with an FNO processor. However, since GNO acts as low-pass filter (Laplacian smoothing) (Li et al., 2018), which creates an information bottleneck for decoding. The GNO decoder receives only the low-frequency output from the FNO processor, making it structurally difficult to reconstruct the high-frequency details essential for high-fidelity generative modeling, which is supported by our experiments (sce Appendix
Comparison with UPT. The UPT model uses GNN for super-node pooling, choice we identify suboptimal for operator learning for three primary reasons. First, as the input discretization becomes finer, the GNN converges to a pointwise operator, which decouples it from the discretization convergence property required of a neural operator. Second, GNN can also be viewed as a low pass filter in the generative tasks, which has the same issue of GNO. Third, UPT relies on random subsampling to define super-nodes is ill-suited for adaptive meshes common in PDE tasks, where important regions with low grid density may be undersampled. For instance, a dense mesh of the Farth, random subsampling would likely over-concentrate super-nodes in the polar regions that of less interest. Furthermore, UPT's decoder uses single-layer, perceiver-style (Jaegle et al. 2022) cross-attention where the queries are derived only from the output positions, which struggles to reconstruct the high-frequener information (Dong et al. 2021). We found that for generative tasks, it is crucial to use a combination of the input function and its position as the query to the cross attention decoder and to use multiple attention layers as shown in the ablation study (Appendix In Table we present a detailed architectural comparison between MINO and GINO/UPT.
Comparison with Transolver. Key differences lie in the receptive field and latent-space design MINO. Transolver employs purely global receptive field, whereas our architecture integrates both local and global information processing. In addition, MINO follows the modern, highly scalable, and efficient DiT paradigm by replacing the grid-dependent "patchify" operation with domain-agnostic GNO encoder. Finally, because the slicing and de-slicing operations in Transolver act on every point, and these operations happen in every Transolver layer, which can be inefficient. In contrast, MINO performs encoding and decoding only once and can be viewed as a reducod-order operator; its decoder is shallow, consisting of one cross-attention block followed by a single self-allention block. As shown in our experiments, MINO is markedly more efficient than Transolver in both training and inference.
C.3 Designing neural operators with general deep learning architectures
The composition of multiple neural operators remains valid neural operator (Wann et al. 2025 Previous works such GINO and UPT adopt the standard encoder processor decoder paradigm. However, directly replacing the processor with a generic neural network is sub-optimal: it forces the decoder to rely solely on a fixed-size latent representation, whereas the desired output lives in an infiniteddimensional function space. To overcome this limitation, we design the decoder via cross attention to take both the latent representation and the original input function as inputs, thereby circumventing the bottleneck imposed by a fixed latent shape.
D Additional Results
In this part, we present additional qualitative results to further demonstrate the performance of MINO compared to bascline models. As shown in Figure 5 MINO generates rcalistic samples for the Navier-Stokes benchmark that closely match the ground truth. In contrast, both GINO and UPT capture only the general, low-frequency trends of the solution, failing to reconstruct the high-frequency information essential for high-fidelity generation. This difficulty ariscs because generative models must learn a velocity field with correct high-frequency components to successfully transform high-frequency Gaussian noise into structured data samples. We provide further qualitative results for MTNO the Darcy Flow, Shallow Water, and
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Published in Transactions on Machine Learning Research (10/2025)
Table 8: Architectural Comparison: MINO vs. GLNO / UPP
Feature
MINO (Mesh-Informed Neural Op- erator)
GINO / /UPT Frameworks
Primary Goal
Probabilistic
Functional Generative
Deterministic
Physics Simulation:
Modeling (learning distributions via flow matching).
GINO solving PDEs varying geome-
tries; UPT- scalable temporal simulation (latent rollouts).
Encoder Strategy
Learned Resampling onto a Grid: a Graph Neural Operator (GNO) projects from Ni. irregular points to a fixed, regu- lar Node grid.
GINO: GNO maps points to a regu- lar grid and augments them with geo- metric features (signed-distance function). UPT: GNN pools information from Ni. points onto a smaller, fixed set of "super- nodes." "
Latent Representation Structured Feature Map (Grid): a tensor of shape (H x W C), compatible to CNNs.
GINO: structured feature map (grid). UPT: unstructured set of vectors (tokens) of shape (Niokena x Duidden).
Core Processor
SOTA diffusion-style U-Net (or Trans- former).
GINO: Fourier Neural Operator (FNO) for efficient global processing on the latent grid.
UPT: Transformer (Approximator) for modeling temporal transitions between la- tent tokens.
Decoder Mechanism
Cross-Attention with Input Bypass: Query (Q) comes from the original input function fi and its positions; Key/ Value (KV) come from the processed latent state. Prevents information bottleneck and preserves high-frequency details.
No Input Bypass (Information Bot- tleneck): GINO projects directly from latent grid to query points. UPT uses Perceiver-style cross-attention where Q is derived from outpul positions only. All information must be reconstructed solely from the compressed latent state.
Uncertainty
Inherent & Computationally Cheap: being generative, the model learns dis- tribution; Monte-Carlo dropout can pro- vide epistemic uncertainty with single trained network.
Computationally Expensive: uncer- tainty is not a primary target; estimating usually requires training an ensemble of models (multiple runs with different. seeds).
Mcsh-GP benchmarks in Figures and respectively, which visually confirm the effectiveness of our approach.
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Published in Transactions on Machine Learning Research (10/2025)
Ground
GINO
Figure 5: Generation of Navier-Stokes samples under different baselines
Ground Truch
MINOT
MINO U
Figure 6: Generation of Darcy-Fil Flocy-Flow samples
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Published in Transactions on Machine Learning Research (10/2025)
2.0
05
Figure 7: Generation of Shallow Water samples
Ground
MINO-T
Figure 8: Generation of Mesh GP samples
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Published in Transactions on Machine Learning Research (10/2025)
E
Ablation Study
We conduct an ablation study to validate the effectiveness of the specific cross-altention mechanisms used in the encoder and decoder of our MINO architecture. We take the standard MINO-T as our baseline and evaluate its performance against two architectural variants:
We replace the cross-attention mechanism in the encoder with a standard self-attention mechanism. In this setup, the Key-Value (KV) pair for each attention block is derived from the output of the previous block, rather than being fixed as the initial latent reprosentation from the GNO.
Self- Aeff-Attention Encoder:
We modify the decoder's query to be derived only from the function's observation positions, similar to the decoder design in prior works like UPT, GINO. This removes the explicit dependency on the input function's values in the query.
Position-Only Decoder:
The results of this study, presented in Table 12 confirm that our proposed design choices are critical for achieving high performance. The standard MINO-T consistently and significantly outperforms both ablated variants across all datasets and metrics. The performance degradation is most severe in the Position- Onlyy Decoder variant. By relying only on positional queries, the dccoder struggles to reconstruct fine-grained details. This result highlights the importance of conditioning the decoder's queries on the input. function's 's values for high-fidelity generative tasks. The Self-Attention Encoder variant also performs worse than standard model, which suggests that using the initial latent representation as fixed context for iterative refinement in the encoder is a more effective strategy than a standard self-attention approach.
Table 9: Ablation study on key architectural components of MINO across four benchmarks. We compare our standard MINO-T against two variants: one replacing the encoder's cross-attention with self-attention, and another using a simpler position-only decoder query similar to prior work like UPT. Lower SWD/MMB scores are better; best results are in bold
Datasct >
Navier Stokes
Shallow Water
Darcy Flow
Cylinder Flow
Variants Metric SWD
MMD
SWD
MMD
SWD
MMD
SWD
MMD
Standard MINO-T
4.0
3.6. 10
9.8
8.7.10
8.9
3.4-10-*
2.9
2.6 10
Self- Attention Encoder 5.2-10-2
10-2
1.7 10-2
10-2
2.0 200110-1
3.7 3.7 10-
10-2
Position-Only Decoder 10-1
10-1
80-1
5.7-10-1
10-1
44-10-1
70-1
50-
F Empirical Analysis of Scaling and Stability
To evaluate performance consistency, we train both MINO-U and MINO-T with cight different random seeds on the Navier-Stokes and Cylinder-Flow datasets. The results, summarized in Table I0 show that MINO-U consistently outperforms MINO-T across both datasets and metrics. We also note that performance variance across seeds slightly higher on the Navier-Stokes task compared to the Cylinder-Flows task.
We next study data-scaling on Navier Stokes benchmark by training on progressively larger subsets of the full 30 k-sample dataset and evaluating on the same test split. Table [[ shows steady performance gains as the training set grows, with improvements tapering off once roughly k samples are reached.
Table 10: Performance comparison over cight random soeds. We report the mean + standard deviation (in the parentheses) for each metric. Lower is better.
Dataset
Navier Stokes
Cylinder Flow
Model Metric +
SWD
MMD
SWD
MMD
MINO-T
3.7 10 (6.8 10 3) 3.1 10
(6.0 10 10-3)
2.9 10
(2.4 10 10-3)
2.6 10 (3.0 10-3)
MINO-U
3.3 10-2 (6.3 10 10-3)
2.7 10-2
2.--10-2 (2.0 10-3)
1.9 10-2 (2.8 10-3)
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Published in Transactions on Machine Learning Research (10/2025)
Table 11: Performance of MINO-U when trained on a subset of the Navier-Stokes dataset.
Metrics Samples +
6,000
12,000
18,000
24,000
30,000
SWD
10-2
3.9 10-2
10-2
3.0 10
2.8 10
MMD
4.0-10
3.2 10
2.2-10-2
3.1 8.1-10 11-110-2
1.9 10
G Comparison with Transolver under Identical Settings
To ensure a fair comparison with Transolver, we reduce the latent dimension MINO from 256 to 192, and use the same training configuration (200 epochs, batch size 48). This results in smaller (13.5M parameters) and MINO-T (S) (12.1M) compared to Transolver (15.0M). Notably, our models are significantly faster, achieving 3.1x speedup in training and 3.5x speedup in inference while maintaining superior performances.
(S)
Table 12: Performance comparison with Transolver under identical settings. Best Performance in bold
Dataset
Navier Stokes
Mesh GP
Cylinder Flow
Model &. Metric
SWD
MMD
SWD
MMD
SWD
MMD
Transolver
5.3-10-2
5.1-10
9.1 10
40-
2.5-10-
2.4 10-2
MINO-T (S)
10-2
4.0-10
7.9-10-2
2.8 10-2
2.7-10-2
2.6 10-2
MINO-U (S)
10
2.5-10-2
6.1-10-2
3.0 10-2
2.5-10-2
2.3-10-2
23


Reviews:
[{"review_id": "lSdV7lUaEM", "reviewer": "Reviewer_3VJC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n- The proposed method effectively integrates the benefits of several prior approaches and demonstrates strong performance across multiple benchmark datasets.\n\n- The paper introduces Sliced Wasserstein Distance (SWD) and Maximum Mean Discrepancy (MMD) as evaluation metrics, which enable fair and comparable assessment across different datasets.\n\n- The writing is clear and well-structured, with a thorough and detailed summary of related work.\n\nWeaknesses\n\n- The comparisons in this paper primarily focus on training-based methods. Including traditional numerical or physics-based baselines would strengthen the evaluation and highlight the practical advantages of the proposed method.\n\n- The Transolver paper reports results on datasets with mesh sizes up to 32,000, whereas this paper only evaluates up to 4,096. This raises concerns about the scalability of the proposed method to significantly larger meshes.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a novel architecture for functional generative models: the Mesh-Informed Neural Operator (MINO). By integrating the strengths of previous methods, MINO effectively captures both local and global receptive fields, handles irregular grids, and learns both low- and high-frequency features. The results show that MINO outperforms existing learning-based methods on both regular and irregular grids.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- In Table 2, are there traditional (non–training-based) methods commonly used for this task? If so, could you include comparisons with such approaches to better position your method?\n\n- Could you also provide experiments on datasets with larger mesh sizes, in order to more convincingly demonstrate the scalability of your approach?", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "U8idbvzeih", "reviewer": "Reviewer_4dSS", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n - The work is well written and easy to follow.\n - The architecture is presented clearly, and the motivation for the various modules (encoder, processor and decoder) is well-motivated\n - The numerical experiments show that MINO has strong performance across various challenging test problems\n - The proposed functional generative evaluation metrics are likely to be of general interest, independently of MINO, which further strengthens the work.\n\nWeaknesses:\n - The work contains a few minor typos. Please see the requested changes section", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this work, the authors propose a novel architecture Mesh-Informed Neural Operators (MINO) which extents existing Neural Operator frameworks (in particular, Fourier Neural Operators) for functional generative models. MINO does not needed a rectangular grid nor a rectangular domain. MINO is still discretization agnostic, and so allows for the application of Neural Operators to new application areas. The authors also introduce two new functional generative evaluation metrics, one based on the sliced Wasserstein distance the other maximum mean discrepancy, to test MINO against existing architectures. Through a convincing set of experiments, the authors demonstrate that MINO achieves state-of-the-art performance on various benchmarks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Typos\n - In Eqn. 3, should the plus sign be concatenation, as in Eqn. 6?\n - The first sentence of Section 4, \"baselines on a suite of functional generative benchmarks under OFM (Shi et al., 2025) paradigm\" should read \"baselines on a suite of functional generative benchmarks under **the** OFM (Shi et al., 2025) paradigm.\"\n - A complaint about the formatting/standards. Many references are given as references to the arxiv version of the work, rather than the published version of the work. As one (of many) example, Ashish Vaswani et al \"Attention Is All You Need\" was published in neurips 2017, yet the references in this manuscript is to the arxiv version of the paper. I think the authors should double-check their references to make sure that the references to the works are to the published works when possible.\n - On page 17, the fourth line \"That is, if we draw anthor new function samples...\" This sentence has a typo.\n - Also page 17, fourth line in the first paragraph of section C.1, \"conditions to a final solution .\" There is an extra space. It should be \"conditions to a final solution.\"", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wE4ZXcDsSt", "reviewer": "Reviewer_fkny", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n\n- the approach is will motivated and explained in it's functionality\n- very clever remix of GINO & UPT (and I mean this as a confidence) , with modifications that I can understand alleviating the oversmoothing issues of GINO and UPT (it might be worth adding https://arxiv.org/abs/2206.10991 and https://arxiv.org/abs/2103.03404 as references explaining the effects in both GNNs and attention)\n- systematic evaluation and ablations\n\nWeaknesses\n\n- single seeds reported without attempts at uncertainty quantifification \n- mildly unfair training/evaluation (different batchsize for transolver, different parameter counts) \n- while the method is well explained for _understanding_ its _distinction_ could be improved (see below)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents MINO (mesh informed neural operator) an improvment of GINO (geometry informed neural operator).\n\nThe key differences are\n\n- different encoder/decoder (incorporating a skip connection from input to decoder and adding cross attention to both en- and decoder)\n- switch in training paradigm (flow matching for learning distributions of operators vs. deterministic learning of PDEs/unrolled PDEs)\n- leveraging the optimizations and training tricks of SotA image diffusion models with a UNET/transformer latent processor\n\nwhich are evaluated on a set of regular and irregular grid datasets, showing improvements across the board", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "# critical\n\n-  perform bootstrapping + a suitable statistical significance test on the test set as a cheap way to add statistical significance (w.r.t to the dataset) \n\n\n# improvement\n## clarify what is mean by \"systematic evaluation of modern architectures for functional generation\"\n\nDid prior work not to modern architectures? not systematic? not functional?  (this one is a nit pick, but one of my triggers)\n\n## train multiple seeds and report mean and std\nIn the same experiment, reduce parameter count of MINO to matchthe count of a transolver such that transolver can fit into the GPU memory (consider checkpointing, mixed precision training etc.) to make a fair comparison (could be worth doing a mini-scaling study on a subset/coreset of the datasets as well, but that's already a bonus idea)\n\n## Concise delta w.r.t to GINO and UPT\n\nI struggled a bit to find the concrete distinctions between GINO and this paper,the below table represents my best attempt after a close reading of the paper + the references. I suggest to add something like this to the appendix or main body\n\n**Architectural Comparison: MINO vs. GINO / UPT**\n\n| Feature | **MINO (Mesh-Informed Neural Operator)** | **GINO / UPT Frameworks** |\n| :--- | :--- | :--- |\n| **Primary Goal** | Probabilistic **Functional Generative Modeling** (learning distributions via flow matching). | Deterministic **Physical Simulation**:  **GINO:** Solving PDEs on varying geometries.  **UPT:** Scalable temporal simulation (latent rollouts). |\n| **Encoder Strategy** | **Learned Resampling onto a Grid:** GNO projects from $N_{in}$ irregular points to a fixed, regular $N_{node}$ grid. | **GINO:** GNO maps points to a regular grid, augmented with geometric features (SDF).  **UPT:** GNN (not GNO) **pools** information from $N_{in}$ points onto a smaller, fixed set of $S$ \"supernodes\". |\n| **Latent Representation**| **Structured Feature Map** (`Grid`): A tensor of shape ($H \\times W \\times C$), compatible with CNNs. |  **GINO:** Structured feature map (`Grid`).  **UPT:** Unstructured set of vectors (`Tokens`) of shape ($N_{tokens} \\times D_{hidden}$). |\n| **Core Processor**| SotA Diffusion-style U-Net (or Transformer)   | **GINO:** Fourier Neural Operator (FNO) for efficient, global processing on the latent grid. **UPT:** Transformer (`Approximator`) for modeling temporal state transitions between latent tokens. |\n| **Decoder Mechanism** | Cross-Attention **with Input Bypass:**  **Query ($Q$):** Derived from **original input function $f_t$** and its positions. • **Key/Value ($KV$):** Derived from the processed latent state. This prevents an information bottleneck, preserving high-frequency details. | **No Input Bypass (Information Bottleneck):** **GINO:** projects from the latent grid to query points.  **UPT:** Perceiver-style Cross-Attention where **Query ($Q$)** is derived from **output positions only**. All information must be reconstructed solely from the compressed latent state. |\n| **Uncertainty** | **Inherent & Computationally Cheap:** As a generative model, it learns a distribution. Can leverage **Monte Carlo Dropout** to estimate epistemic uncertainty from a single trained model. | **Computationally Expensive:** Not a primary goal. Requires training a full **ensemble of models** (i.e., multiple runs with different seeds) to estimate model uncertainty. |", "broader_impact_concerns": "Not applicable/sufficient", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)