GridMix: Exploring Spatial Modulation for Neural Fields in PDE Modeling
Authors: Honghui Wang
Shiji Song
Gao Huang
Venue: ICLR 2025 Oral (2025)
Keywords: Partial Differential Equations
Neural Fields
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Significant advancements have been achieved in PDE modeling using neural fields. Despite their effectiveness, existing methods rely on global modulation, limiting their ability to reconstruct local details. While spatial modulation with vanilla grid-based representations offers a promising alternative, it struggles with inadequate global information modeling and over-fitting to the training spatial domain. To address these challenges, we propose GridMix, a novel approach that models spatial modulation as a mixture of grid-based representations. GridMix effectively explores global structures while preserving locality for fine-grained modulation. Furthermore, we introduce spatial domain augmentation to enhance the robustness of the modulated neural fields against spatial domain variations. 
With all these innovations,
our comprehensive approach culminates in MARBLE, a framework that significantly advancing the capabilities of neural fields in PDE modeling. The effectiveness of MARBLE is extensively
validated on diverse benchmarks encompassing dynamics modeling and geometric prediction.

Published as a conference paper at ICLR 2025
GRIDMIX: EXPLORING SPATIAL MODULATION FOR NEURAL FIELDS IN PDE MODELING
Honghui Wang, Shiji Song & Gao Huang*
Department of Automation, BNRist, Tsinghua University, Beijing, China, wanghh20@mai 1s. tsinghua edu. cn, {shifis, gachuang) Stsiannuaatsiggg edu.cr
ABSTRACT
Significant advancements have been achieved in PDE modeling using neural fields. Despite their effectiveness, existing methods rely global modulation, limiting their ability to reconstruct local details. While spatial modulation with vanilla grid-based representations offers promising alternative, it struggles with inadequate global information modeling and overfitting the training spatial do- main. To address these challenges, we propose GridMix, novel approach that models spatial modulation as a mixture of grid-based representations. GridMix effectively explores global structures while preserving locality for fine-grained modulation. Furthermore, we introduce spatial domain augmentation to enhance the robustness of the modulated neural fields against spatial domain variations. With all these innovations, our comprehensive approach culminates in MARBLE. framework that significantly advancing the capabilities of neural fields in PDE modeling. The effectiveness of MARBLE is extensively validated on diverse benchmarks encompassing dynamics modeling and geometric prediction. The code will be available on httts//ggthbbbccco//c/ccaottuttt
INTRODUCTION
In recent years, deep learning has attracted considerable interest in its ability solve partial differ- ential equations (PDEs), providing data- driven approaches to approximate solutions across broad spectrum of problems (Lu al. 2021 Li et al., 2021; Brandstetter al., 2022). One particularly innovative direction within this field is the use of neural fields, or implicit neural representations
(INRs) Park et al. 2019 Sitzmann et al. 2020), which offer continuous parameterizations of PDE solutions, enabling high-resolution modeling and flexible predictions across diverse geometries.
Building upon this foundation, INRebased methods (Yin et al.) 2022 Serrano et al. 2023) have demonstrated notable advancements in PDE modeling. These methods leverage INRs to encode solutions-exxe expressed as functions of spatial coordinates-into a low-dimensional latent space via auto-decoding (Park al.) 2019). The solution operator is then approximated by learning how these latent representations evolve over time or in response to varying conditions. This framework effectively bridges the gap between complex spatial domains and the underlying PDEs, achieving impressive efficiency and accuracy in both steady-state and transient scenarios.
Despite their potential, INReb methods face limitations in accurately representing complex spa- tial variations in PDE solutions, often missing fine-gras ine-grained local details. This issue stems from the use of global modulation (Perez al. 2018), a technique that allows base INR represent differ- ent functions by altering its behavior conditioned on function-specific latent representations. Specif- ically, these latent representations are utilized create function- funetion-specific modulation parameters that modify the hidden activations of INRs, thereby altering the output. As these parameters are typically shared across all spatial positions, the model's ability learn detailed local features and variations is hindered (Bauer et al. 2023). Additionally, INRs are susceptible to spectral bias (Rahaman al.) 2019), which can restrict their capability to learn nigh-freeqency components of solutions.
To address the limitations of global modulation, we propose extending it to spatial modulation. drawing inspiration from the success of grid-based representations in computer vision (Liu et al.)
"Corresponding author

---

Published as a conference paper at ICLR 2025
2020 Müller et al., 2022) Grid-basee representations discretize continuous space into regular grid, associating each grid point with a learnable vector that encodes local features. By applying this concept to PDE modeling, enabling a more fine-graid fine-grained representation. However, while grid-based representations have proven effective in mitigating spectral bias in vision tasks under the assumption of fixed and regular spatial domain, their straightforward adaptation to spatial modulation for PDE modeling presents new challenges. To be precise, although using independent modulation parameters at each location excels at capturing local information, it often compromises the ability to model global structures in sparse or irregular domains, resulting in poor reconstruction quality in unseen regions. This trade-off limits their generalization across varying spatial domains, which is crucial for robust PDE modeling.
can assign grid-based modulation parameters to each function,
In this paper, we introduce GridMix, a novel spatial modulation technique that builds on the strengths of grid-based representations while tackling the aforementioned challenges. We achieve this by rep- resenting spatial modulation as a mixture of basis functions, where each is parameterized through grid-based representations. GridMix preserves the fine-grained locality of grid-based modulation, allowing for detailed exploitation of local information in function reconstruction. At the same time, it regularizes the modulation space to the linear span of small set of learned grid basis functions, which are shared across function instances and facilitate the extraction of global structural informa- tion. This regularization effectively mitigates the risk of overfitting to specific spatial domains. Ad- ditionally, we introduce Spatial Domain Augmentation, which simulates domain variations during training to further enhance generalization to unseen regions. Together, these innovations form the GridMix Augmented Coordinate-basee Neural Fields (MARBLE) for PDE modeling, significantly improving the learning capacity of previous approaches that relied on global modulation.
Our contributions are summarized as follows:
1. We introduce GridMix to mitigate the spatial domain over-fitting of spatial modulation.
2. We present the MARBLE framework, which combines GridMix with spatial domain aug- mentation to boost the performance of existing INR-based methods for PDE modeling.
3. We demonstrate the effectiveness and versatility of MARBLE with extensive experiments, including dynamics modeling and geometric prediction.
2 RELATED WORK
Neural PDE Solvers. Deep learning has rapidly advanced as a powerful tool for solving Partial
Differential Equations (PDEs) across diverse fields, including fluid dynamics (Kochkov al.) 2021
Sun et al. 2023), solid mechanics (Samaniego et al. 2020 Nguyen- Thanh et al. 2020). and inverse
problems (Lu et al. 2022 Molinaro et al.) 2023). Neural Operators have emerged as a leading approach within this domain, modeling the solution operator between input and output function
spaces (Lu et al. 2021 Li et al. 2021). Recent research has centered on addressing key challenges for Neural PDE solvers, such handling intricate geometries and ensuring precise predictions. To tackle these issues, methods like Geo-FNO et al. 2023b) and MP-PDE (Brandstetter et al. 2022)
have been developed to handle irregular grids, while Factorized-FNo (Tran et al. 2023 refines FNO's architecture and training strategy for improved performance.
Neural Fields, also known as implicit neural representations (INRs), utilize neural networks (typ- ically MLPs) represent continuous signals based on spatial coordinates. These models have demonstrated exceptional performance in various vision tasks, including image representation (Sitz-
mann al. 2020; Dupont al. 20022). 3D shape modeling (Park al. 2019 Tancik et al. 2020),
and novel view synthesis (Mildenhall et al.) (021). In the realm of PDE solving, INRs have been integrated into dryysics-informer neural networks (PINNs) (Raissi et al., 2019: Wang et al. 2023 2024) to approximate solutions. While effective, traditional PINNs require training a separate model for each unique set of initial and boundary conditions. To address this limitation, recent research has explored modulating the intermediate features of base INRs using instance-specific latent codes, enabling the representation a solution space (Yin et al. 2022 Serrano et al. 2023). These INR-based approaches are independent to spatial discretization and scale well to high-dimensional problems. However, INRs are susceptible to spectral bias (Rahaman et al.) 2019). preferentially learning low-frequency components, which can hinder their representational capacity.
2

---

Published as a conference paper at ICLR 2025
Grid-based Representations offer compelling solution to the aforementioned limitation of INRs (Liu et al. 2020 Takikawa et al.) 202T; Müller et al. 2022 Lee et al. 2024b). These ap- proaches discretize the input domain into tinegrained gnd. assigning learnable feature vector to each grid cell. When querying points within this space, local feature vectors are aggregated with interpolation and integrated into INRs, either as input (Mäller et al. 2022) within intermediate layers (Lee et al.) (2024b). The inherent locality of grid-based representations enables INRs to ef- fectively capture high-frequency signals. While existing grid-based methods excel at reconstructing fine details within fixed domains, their generalization to unseen regions is restricted, hindering their applicability to PDE solving. To address this challenge, we propose learning grid features as mix- tures of basis representations, promoting domain generalization while preserving locality. A related approach, Factor Fields (Chen et al. 2023), decomposes signals into components like coefficient fields and basis functions. However, while Factor Fields operates in the signal space, our work applics a similar decomposition in the modulation space. Additionally, Factor Fields focuses on tasks outside the PDE domain, whereas our approach targets PDE-related challenges, emphasizing domain generalization of spatial modulations.
METHOD
We introduce MARBLE (GirdMix Augmented Coordinate- Neural Fields) for PDE modeling. We begin with the problem formulation in Section 3.1 Next, we provide a preliminary overview of CORAL (Serrano al. 2023), INR-based method for solving PDEs, in Section 3.2 Finally, describe the spatial modulation with MARBLE and the spatial domain augmentation in Section 3.3
3.1 PROBLEM DESCRIPTION
We focus PDE modeling tasks that involve approximating the operator G", which maps functions from input space A C L (2,, d)) to output space U C (S2, _ according to the governing PDE. Here (2,, Rddx) denotes the infinite-dimensioaal space of square-integrable functions with domain St and range in Rdd,, Two such exemplary tasks are investigated: 1) Dynamics Modeling: Here, the objective is capture the temporal evolution of physical system over a forecasting horizon. This translates to modeling the transition from state to state Ut+ót, where öt represents the time step. 2) Geometric Prediction: This task involves making predictions based solely on the geometric configuration of the system. In the context of geometric prediction task, each data sample is observed on a unique domain Xg Conversely, for modeling system dynamics, a single domain Xer is employed for training the model across all examples, while separate domain Xze is used for evaluating the model's performance during testing. Xe, and Mee are subsets a full domain K
3.2 PRELIMINARY
The CORAL Framework addresses PDE modeling tasks through two-stage training. In the first stage, the reconstruction stage, CORAL uses two modulated INRs, fe..s. and fe..... to parametrize input and output functions, respectively. Each of these INRs serves base model that is modu- lated with function-specific parameters to reconstruct different functions. Specifically, INR param- eters O2 and O,, are shared across all functions within their respective space, while the modulation parameters, Oo; and Ouz, are unique to each function. The modulation parameters are derived from low-dinensional latent codes za; and 2u, through hypernetworks h2 and hy: Pa. = ha (Zo.) and Φ.. = h ). After training. each function is represented by a compact latent code. with the reconstruction error indicating the quality of the representation. In the second stage, the forecasting stage, CORAL learns mapping between latent codes 2a, and zu, using processing network Jø-
During inference, CORAL operates in three steps, as shown in Figure First, the encoder ea A H> Rdd projects input a; into latent space (Za,) ) via auto-decoding (Park et al. 2019), as illustrated in Tigure[ib, Given a base INR fa_.... ea encodes the function a; by optimizing the latent code zn, to minimize the reconstruction error:
Lx. (Fo.. 0.) Ex [o.., (a) (x))) where da; = ha (a(s))
Next, the model gø RRd + Rd transforms &a, into the output latent code zu,- Finally, the decoder E. : IRd, + 24 decodes the processed code back into the output function space through a single forward pass, as shown in Figure
3

---

Published as a conference paper at ICLR 2025
Input Space A
Output Space U
(x())
Update Zu, to minimize Eq.
fa
ha
Modulared INR fa.
Modulated INR fe.
Modulation C
Zai
Xtr
backward forward
(h) Encoding
(x(*))
Processing
feu
g
x Modulation Qu.
Encoding
Decoding
(a) CORAL framework
(c) Decoding
Figure 1: INR- framework for PDE modeling.
Meta-Learned INRs. Typically, the encoding process could require thousands steps gradient descent. To reduce this fitting cost, CORAL adopts the Model-Agnostic Meta Learning (MAML) algorithm (Finn et al. (2117) to meta-learn the parameters 0, such that the encoding process can be done in a few gradient steps (Dupont al. (022). More specifically, a step of the MAML inner loop optimizes a latent code Za, to fit randomly sampled function a; as:
0; Ro.
(k+1)
- av (1) Paz
Lxi, ( 100,0, ai), with Po. ha ha (k) for S k K-1,, (2)
where is the inner-loop learning rate and K the number of gradient steps. The outer loop then meta-learn the shared parameter 0g by
0₂ - Ou ADD En Lxx,
az),
where / denotes the outer loop learning rate and N the number of samples.
Global Modulation in CORAL. CORAL utilizes SIRENs (Sitzmann et al. 2020) as the backbone for its INRs. SIRENs are multilayer perceptrons with sine activations, which can be formulized
(a) (L(oL-1 (00--0....00(ss))) oo)(1)) b2, with (v)())) sin(wo W (x)) + b.)). (4)
where 0 = (W,, fi) are network weights and biases, Wp is positive scaling factor, 70(x) = and (2) are activations of the i-th hidden layer given coordinate æ. CORAL applys shift modula- tions (Perez et al. 2018) to SIRENs to represent individual function as
(x) =
000 +x)) with (1)())) = sin sin(wo(Wn) (Z) +:)), (5)
where o = (=) represents modulation parameters at the i-th layer. Note that the global modula-
tions Φ; shared across different spatial coordinates shown in Figure a). This property limits
the capacity of modulated INRs to represent complex function space. As demonstrated in previous work (Bauer al. 2023), this global modulation fails capture local details because any changes in modulations will lead to global perturbations across the reconstructed function.
3.3 SPATIAL MODULATION WITH GRIDMIX
While the original modulation design only incorporates global information of individual function to adjust the behavior of the based INR, propose enhanced approach which leverages local infor- mation for fine-grained modulation, without compromising its ability to utilize global information.
Spatial Modulation Framework. Building on prior work (Müller et al. 2022 Lee et al. 2024b)
that highlights the effectiveness of grid-based representations in enhance the learning capacity of INRs, we extend the global modulation scheme to a spatial modulation approach (Figure (a b)). We achieve this by introducing single-channel grid- representation &i RR Hx where H and

---

Published as a conference paper at ICLR 2025
N
Spen
+
P
(a) Global Modulation
(b) Spatial Modulation
(c) GridMix (Ours)
Figure 2: Different modulation strategies. We visualize the hidden activations and modulation pa- rameters as blue and green tensors, respectively. The hidden activations has dimensions N x C, where N represents the number of query locations and C' the number of channels. The green arrow indicates the dimension along which the modulation parameters are broadcast.
W are spatial resolution of the grid (considering a 2D domain an example). To derive the position- dependent shift modulation scalar (x) for given spatial location a from this grid, we first identify the neighhoring grid points surrounding this location (in 2D, this involves four points). We then
compute oi (2) by interpolating the values at these neighboring points, using bilinear interpolation in the 2D case. The spatial modulation reformulates Equation as
oi (T)()) = (Woo(WWW()))) + b, ((())))
(6)
By assigning learnable parameters to each grid cell, this approach enables the model to effectively capture and adapt to spatially varying features.
While the localized nature of spatial modulation enhances learning capacity, it also introduces limi- tations, particularly in capturing global information. The flexibility of grid-based modulation allows meta-learned INRs to minimize the reconstruction error Lx,, (fo_.da; a.) independently at each location x E Xy. However, this localized focus can overlook broader contextual relationships es- sential for comprehensive understanding of the data. As a result, the model may become overly specialized to the training domain Me,, leading to over- over-fitting. This over-fitting impairs the model's ability to generalize effectively to new data in unseen test domains Xte.
To further illustrate this limi- tation, we compare the perfor- mance of INRs with global and local modulations under two set- tings: (1) the test function is ob- served on the training domain Ner and reconstructed on &ou and (2) the function observed on test domain X The results in Table D and Figure [ clearly demonstrate that, although spa- tial modulation significantly re- duces reconstruction error Xy,, it fails to capture the global information of test data, result- ing in a large error on Xfull- Moreover, it performs poorly
Table 1: Comparing different modulations Navier-Stokes with irregular grid (π = 20%). We use the data on observed domain to optimize the modulation parameters via Eq. ( and evaluate the reconstruction error on inference domain with the optimized modulations. Metries in MSE.
Observed Domain
X
Inference Domain
Nr.
Xysit
N,.
Global Modulation
1.32c-4
1.45e4
3,48c-3
3.72c-3
Spatial Modulation
3,95e-5
5.78e-2
4.08e-1
4.67e-1
Single-channel GridMix
3.17e-3
3.88c-3
9.56c-3
9.96c-3
Multi-channel GridMix
1.90e-5
2.85e-5
1.94e-3
2.25e-3
Global Modulation + SDA
2.88c-4
3.15e-
4.50e-4
4.85e-4
Multi-channel GridMix SDA
7.45e55
9.64e-5
1.38c-4
1.49c-4
with data observed on unseen domain in Xze. While this limitation is less critical in vision tasks, where fixed and regular spatial domain typically assumed, it becomes a significant concern in the context of PDE modeling, where generalization across varying spatial domains is crucial metric.
GridMix. To mitigate over-fitting associated with spatial modulation with respect to the training spatial domain, we introduce a mixture of grid-based representations for generating spatial modula- tions, illustrated in Figure ce.. We begin by defining set of grid-based representations serve as basis functions and construct the spatial modulations as a linear span of these basis functions. Specifically, the grid mixture in each hidden layer given by:
(x(x)) M C T (x), where
(c1, C2 - CL-1)= h(2) Wn + ba-
5

---

Published as a conference paper at ICLR 2025
Reconstruction on Ass
on
Reconstruction
Aog
Mod. Modulation
Modklasion
Global
Ioboc Jelion
Spatial
GirdMit
(b) Test data is observed on Nee Figure 3: Comparison between INRs with global modulation (Top), spatial modulation with vanilla grid-based representation (Middle) and GridMix (Bottom). Each model is trained with data observed on Nrr. During inference, test data is observed on (a) Xxr (b) Ute and reconstructed on A yaal-
(a) Test data observed on Ker.
Here ( mm=1 denote M grid basis functions, and e; = (c at the i-th layer. These mixture coefficients are efficiently determined from low-dimensional em- bedding& via linear hypernetwork h. The grid basis functions are shared across different function instances and are learned alongside the modulated INRs.
/==1 represent the mixture coefficients
This approach not only maintains the localized nature of grid-based modulation but also enables the capture of global information through the shared basis functions. The rationale behind using grid mixtures parallels the principles of spectral methods, where problem is decomposed into simpler components that can efficiently capture both local and global features. Additionally, GridMix as a form of regularization to the modulation space. For instance, in vanilla spatial modulation, each function is assigned H x W parameters, resulting contrast, GridMix reduces this dimensionality to M, where M represents the number of shared basis functions used across all functions. This strategy effectively prevents the modulated INRs from over-fitting to the data distribution on specific spatial domain, thereby improving the model's ability generalize across a diverse range of previously unseen domains (see Table [ and Figure
high-dimensional parameter space. In
Channel-wise Modulations. While GridMix effectively regularize the modulation space to pre- vent over-fitting, they can also limit the model's representational capacity. To address this issue, we propose incorporating channel-wise modulations into GridMix. Although single-channel approach often performs adequately with vanilla spatial modulation, it falls short within the more constrained framework of GridMix. Therefore, extend single-channel grid basis functions 4tmh #W to multi-channel ones 4." I XX where C' is the number of channels. This extension allows for a spatially dependent modulation vector (z(z), which combines the benefits of GridMix with enhanced expressiveness. Despite this increase in representational capability, the compactness of the modulation space is preserved by sharing the mixture coefficients across all channels.
Spatial Domain Augmentation. While our previous design improves generalization across spatial domains through modulation strategies, we now tackle the problem from different perspective by refining the learning process itself. One inherent reason for the limited generalization is the lack of domain variation during training, as the training domain X,, remains fixed in Equations 2 and To counter this, we propose data augmentation of spatial domains to simulate domain variations encountered during inference. To illustrate, for each run of inner and outer loops, we randomly sample a subset of training coordinates Xsub from the fixed domain X as described below:
Inner loop: Za,
(k+1)
(kk za.
aV Exiaxx (S. Rax
Voa.d (en ai), 0 S K -
(8)
Outer loop: 02 + 0. BV,
(9)

---

Published as a conference paper at ICLR 2025
By incorporating spatial domain augmentation, we enhance the robustness of modulated INRs with respect to the variation of spatial domain.
4 EXPERIMENTS
We assessed the flexibility of our model by applying it to two different tasks: forecasting the tempo- ral dynamics of physical system (see Section 4.1) and predicting the steady-state behavior based on domain geometry (see Section 4.2). Both tasks were conducted using the experimental setup de- scribed in the CORAL paper (Serrano et al., 2023). with additional details available in Appendix B Moreover, a thorough ablation study our method is presented Section 4.3
4.1 DYNAMICS MODELING
The objective of dynamics modeling is to forecast the temporal evolution of a physical system over a specified horizon. To accomplish this, we follow an autoregressive framework operating within a latent space. Formally, given sequence of target functions (uo, Ust..... ..., uT5r), where ue represents the system state at time and öt the time step, our aim is to predict the subsequent states ukst for k 1, ..... based the initial condition M4p- This is achieved by encoding ug into latent represen- tation 2/ = e(uo) and recursively generating subsequent latent codes 2kōt using an autoregressive processor gv- Finally, the predicted latent states decoded to obtain the corresponding estimated functions Us.se = ((Za6t), ), k I....T..
Architectures. For this problem, employ single modulated INR f8.6 represent the func- tion space encompassing the physical quantity across diverse initial conditions and time instances. The modulated INR is trained by the meta-learning algorithm detailed in Section 3 For the au- toregressive processor Jv, a Neural Ordinary Differential Equation (NODE) (Chen et al. 2018 is adopted, enabling flexible predictions at arbitrary time steps. Given an initial latent state 24. the NODE solver recursively computes the latent code at desired time steps (+ according to the equa-
tion St+- Iw 22, 7) 24 1e (= where Çe is neural network parameterized by th Runge-Kutta
The integral term is approximated using a fourth-order Runge- Kutta scheme. Training of the NÓDE formulated the minimization problem:
arg ming. Bu-,,,t- ((o t) e(u
(10)
where Vu denotes the distribution of physical quantity trajectories.
Datasets and Baselines. We generate two datasets of fluid dynamics using the 2D Navier-Stokes equation (Navier-Stokes), which models viscous, incompressible fluid on regular domain, and the 3D spherical Shallow-Wate equation Shtallow-Water which describes the movements of the Earth's 's atmosphere. These datasets consist of time-evolving vorticity fields (and height for Shallow- Water), derived from diverse initial conditions. More details can be found in Appendix [A. Several state-of-the -hrt models let al.) 2021) and FNO (Li et al. )). a mesh-based network (MP-PDE (Brandstetter et al. and two coordinate-based approaches (DINo (Yin al. 2022) and CORAL (Serrano et al.
compared, encompassing two neural operator methods (DeepONet (Lu
2022
2023
The baseline results for comparison are sourced from Serrano et al. 2023).
Evaluation Criteria. To assess the model's spatiotemporal generalization capabilities, two eval- uation protocols are employed. Temporal Extrapolation: Each tracjectory is partitioned into two equal- uujetrajectories of 20 timestamps: an in-time (In-t) and out-of-time (Out-r) seg- ment. The model is trained to forecast up to the end of the In-f segment and evaluated on both horizons. This assesses the model's ''' ability to predict within and beyond the training regime. Spa- tial Subsampling: From the original domain Xyul, with resolution of 64x64 for Navier-Stokes and 64x 128 for Shallow-Water
we randomly select π percent of points as the training grid dey. A distinct grid Xre with identical sparsity is created for testing. Both grids remain constant across different trajectories, allowing for evaluation of the model's ability to generalize to unseen locations.
Results. Table 2 comprehensively evaluate the temporal extrapolation capabilities of various ma- chine learning models on the Navier-Stokes and Shallow- Water datasets under different grid densi- ties (100%, 20%, and 5%). Our method consistently yields the best near-best Mean Squared Error,

---

Published as a conference paper at ICLR 2025
Table Dynamies Modeling Test results. Metrics in MSE. The best results are in bold, while the second-best results are underlined. The "lin. int." abbreviates linear interpolation of irregular grid data onto a regular grid. The term "n.a." indicates "not available", as the Shallow- Water data is in a 2D spherical grid (latitude and longitude), making it incompatible with linear interpolation.
dataset
Navier-Stokes
Shallow- Water
Ner 4 Nic
In-1
Out-1
In-1
Out-1
DeepONet
4.72e-2 1: 2.84e-2
9.58e-2 +: 188e-2
6.54g-3 4.94e-4
8,93e-3 1: 9,42
FNO
5.68c-4 + 7.62e-5
8.95e- +: 1.50e-3
2.51ee
1.17e-4 1: 3,01e-5
100% regular grid
MP-PDE
4.39e.4 +: 5.788
4.46e:3 +: 1.28e-3
5.566-
1.53e:3 +: 2.62e-4
DINo
127c-3++2.200
1.1lc-2
4
2.63e-3 1.36c-4
CORAL
1.86e-4 1.44e-
1,02e-3 8.66ee
3.44e-6 1: 4.01e-7
4.82e-4 1 5.16
MARBLE (Ours)
3.52e-5 5.31e-6
5.04e-4
8.21e-7 = 1.13e-8
1.42e-4 + 7.07e-6
DeepONet
8,37e-1 2.07e-2
7.80e- 1.366-1
1.05e-2 :: 5.01e
1.09e-2 1 6.
FNO + lin. int.
3.97c-3 + 8.03e-4
9,92c=3 2.36c-3
n.a.
n.a. 2.56e-2 1 8.23e-3
20%
MP-PDE
1.3e-11
5.28e-3 1: 5.25e-4
ireegular grid
DINo
9.99e-4 6.7le-3
3.6le-3
2.20e-3 + 1.06e-4
4.94e=3 t 1.92e44
CORAL
2.18e:3 + 6.88e.4
6.67e:3 3.00ee3
1.41e33 1 1.39e-4
2.1 +: 5.58e-5
MARBLE (Ours)
1.62e-4 = 2.42e-5
9.27e-4 1.44e-4
7.06e-4 + 4.60e-5
8,45e-4 + 3.01e-5
DeepONet
7.86e-1 +:: 5.48e-2
7.48e-1 2.76-22
1.11e-2 + 6.94e-4
1.12e-2 +: 7.79c-4
FNO + lin. int.
3.87e-2 + 1.44e-2
5. 1.10e2
n.a. 1.10c-2 # 4.23c-3
n.a. 4,94e-2 + 2.36c-2
5% irregular grid
MP-PDE
1.92c-1 = 9.27e-2
2.17e-1 4.73e-1
DINo
8.65c-2 + 1.16c-2
9.36e-2
1.22e-3 + 2.05e-4
1.52e-2 + 3.74e-4
CORAL
2.44e-2 +: 1.96e-2
4.57e-2 1.78e-2
8.77e-3 1 7.20c-4
1.29e-2 + 1.92e-3
MARBLE (Ours)
1.43e-3 + 5.66e-4
4.73e-3 + +.33e-3
5.92e-3 + 3.32e-44
5,98e-3 t 3.51e-4
substantially surpassing baseline performance in most scenarios. Specifically, at 100% grid density, our method underscores the efficacy of spatial modulation, achieving up to a 5.3x reduction in error for in-time predictions and a 3.3. 3.4x reduction for out-of-time predictions compared to the global mod- ulation baseline, CORAL. Additionally, our method generally outperforms FNO in most settings and matches its performance in the Shallow- Haallow-Water Our-f scenario. Even under the more challenging 20% and 5% grid density settings, our approach retains its superiority, showcasing the effectiveness of GridMix and spatial domain augmentation in mitigating over-fitting to training domain. Notably, on the Kavier-Stokes dataset, it delivers robust performance with up to a 94.1% 1% reduction in In-r MSE and a 89.5% reduction in Out-I MSE compared to the rest-performing baseline. These results highlight the strength of our method in handling dynamics modeling tasks, particularly under sparse and irregular grid, where it demonstrates both accuracy and resilience. Note that DINo outperforms our method in the In-r case of Shallow- Water (π 5%) but underperforms in Out-t. We attribute this to DINo's 's preli-optimiize latent embeddings, which lead over-fitting on training data.
4.2 HHOMETRY-AWARD INFERENCE
This section explores the inference of steady-state system behavior based on domain geometry. For a system with domain Ωg its geometry is discretely represented as a point cloud or structured mesh X, Ωg. This mesh is deformation of reference grid X' to fit the specific object shape, such as an airfoil. The problem is formulated as an operator learning task, where the imput is the grid deformation defined on the reference grid X' and the output is the corresponding physical quantity My defined on Ω2 We aim to develop a model that can generalize to unseen geometries.
Architectures. We utilize two modulated INRs to encode the grid transformations and physical quantities, respectively. A simple MLP is employed as the processor Iø to learn the mapping be- tween the latent codes of these two function spaces.
Datasets and Baselines. To evaluate the proposed method, three benchmark datasets introduced Li al. (2023a) are considered, including the Euler equation WACA-Euuer) Navier-Stokes equa- tion (Pipe) and Hyper-elastic material (Elasticity). More details can be found in Appendix A.2 Our method is compared against three state-of-the tofftthean art models, Geo-FNO (Li et al. 20236). Factorized- FNO (Tran et al.] 2023) and CORAL (Serrano al. 2023), as well two regular-gric baselines, FNO (Li et al. 2021) and UNet (Ronneberger et al., 2013), which are applied to the data after interpolation. The baseline results for comparison sourced from Serrano et al. 2023).
8

---

Published as a conference paper at ICLR 2025
Table Georetry-aware inference Test results. Metrics in relative L2 error. The best results are
shown in bold, while the second-best results are underlined.
Model
NACA-Fuler
Elasticity
Pipe
FNO
3.85e-2 3.15
4.95e-2 +: 1.21e-3
1.53e-2 +:: 8.19c-3
UNet
5.05e-2 1.25c-3
5.34e-2 2.89c-4
2,98e-2 + 1.08e-2
Geo-FNO
1.58e-2 1.77e-3
3,41e-2 = 1.93e-2
6.59e-3 + 4.67e-4
Factorized-FNN -NO
6.20e-3 3.00c-4
1.96e-2 +: 2.00e-2
7.33e-3 + 4.66e-4
CORAL
5.90e-3 1.00c-4
1.67e-2 +: 4.18
1.20e-2 + 8.74e-4
MARBLE (Ours)
5.79e-3 9.67e-5
1.12e-2 + 9.43e-5
1.03e-2 + 2.62e-4
Results. We present our results in Table Notably, our method demonstrates consistent enhance- ment over the baseline method with global modulation across various geometry-aware inference tasks. For instance, MARBLE achieves the lowest error rate of 1.12e-2 on Elasticify, outperforning CORAL by 32.9%. Although it slightly underperforms compared Geo-FNO and Factorized-FFV on Pipe, MARBLE still delivers an improvement of 14.2% over CORAL. These results highlight the effectiveness of GridMix in enhancing generalization to unseen geometries.
4.3 ABLATION STUDY AND ANALYSIS
Effects of Spatial Domain Augmentation and GridMix. By incrementally incorporating the pro- posed mechanisms into the baseline model, CORAL, we observed a consistent improvement in per- formance, as illustrated in Table 4a The spatial domain augmentation notably enhances bascline performance on unseen domains Eze. demonstrating its effectiveness in mitigating over-fitting to the training domain. This approach also proves robust across varying sampling ratios (Table 4b). Building on this, the multi-channel GridMix further elevate the performance, with improvements of 60.7% and 51. .00 for In-1 and Out-I, respectively. These gains align with the regular grid setting. where error rates are reduced by 80.8% and 50.6%.
Table 4: MARBLE ablation experiments on Navier-Stokes. We use the irregular grid setting with 20% assess (a) the effects of core components, including spatial domain augmentation (SDA) and multi-channel GridMix (MCGM). We also examine (b) the impact of SDA's sampling ratio in this setting. Conversely, the regular grid setting is employed to analyze (c.d) GridMix design and (e) latent dimensions. The reported metric is MSE. Default settings are marked in gray
(a) Core components.
(b) Sampling ratio.
SDA
MCGM
In-t
Out-1
ratio
In-t
Out-t
2.18e-3
6.67e-3
0.2
5.94e-4
3.40e-3
4.22e
1.89e-3
1.62e44
9.27e-4
1.62e-4
9.27e-4
0.6
1.67e-4
1.27c-3
(c) Grid resolution.
(d)
Grid basis functions.
(e)
Latent dimension.
In-t
Out-1
num
In-1
Out-1
dim
In-t
Out-
4
4.60e-5
6.10e-4
8
4.02e-5
8.33e-4
16
4,04e-5
5.49e
3.52e-5
5.04e-4
16
3.93e-5
5.36e-4
32
3.52e-5
5.04c-4
3.41e-5
5.78e-4
32
3,52e-5
5.04e-4
64
4,70e-5
6.19.-4
32
4.10e-5
1.68e-3
64
3.84e-5
5.54e-4
128
6.66e-5
8.81e-4
explore the impact of grid resolution and the number of grid basis functions. Our findings indicate that the performance plateaus as resolution and basis numbers increase. We hypothesize that this saturation occurs because, although higher resolution and more basis functions offer greater flexibility in encoding the function space into the latent space, they also lead to increasingly complex latent trajectories that are more challenge to learn.
GridMix Design. Table 4c and 4d
Latent Dimension. We investigate the influence of latent code dimension in Table 4e GridMix constrains the modulation's 's degrees of freedom to the number of mixture coefficients, enabling the use of a smaller latent space dimension. As demonstrated in Table 4e the optimal performance is achieved with a latent dimension of 32. This reduced dimension results in a more compact latent space and simpler latent trajectories for dynamics modeling.

---

Published as a conference paper at ICLR 2025
In
Out
Reference
CORAL
Ours
Figure 4: Visualization of predictions of MARBLE on Shallow- Water dataset with л 20%.
Comparison with Scaled Baselines. To
Table 5: Comparison with scaled baselines Experi-
account for the additional learnable pa- rameters introduced by grid mixtures, we compare MARBLE against scaled base- lines. Specifically, we evaluate CORAL with hidden dimension of 512 (CORAL- 512), which has a comparable number of parameters MARBLE, and CORAL with a hidden dimension of 768 (CORAL- 768), which has approximately twice the number of parameters as MARBLE. As shown in Table the increase in parameters reduces the reconstruction error, leading better fit of the data space. However, this does not necessarily translate to improved dynamics forecast performance. This finding aligns with earlier research by Serrano al. (2023), as outlined in their Table 11. These results highlight the effectiveness of MARBLE in leveraging more parameters to enhance the performance of operator learning within the INR-based framework.
ments are conducted on Navier-Stokes (π 20%) with spatial domain adaptation (SDA). RecErr denotes the test error on the reconstruction stage. Metrics MSE
Model
Params
RecErr
In-1
Out-t
CORAL-128 SDA
83.1K
4.50c-4
4.22c-4
1.89c-3
CORAL-512 + SDA
725.5K
1.69c-4
3.79c-4
1.95c-3
CORAL-768 SDA
1,48155K
1.44c-4
1.17e-3
5.83e-3
MARBLE (Ours)
823.1K
1.38e-4
1.62e-4
9,27e-4
Visualizations. Figure shows the predictions of MARBLE on the irregular grid setting with x 20% for the Shallow- Water dataset. We compare these results with CORAL, the best-performing baseline. Notably, MARBLE excels at modeling complex spatial variations in dynamics forecasting, particularly in the Out-t scenario. Additional visualizations can be found in Appendix C
DISCUSSION AND CONCLUSION
This paper has presented MARBLE, a novel approach for PDE modeling that leverages the power of implicit neural representations with innovative spatial modulation techniques. Our method ad- dresses the limitations of existing INR-based methods by introducing GridMix, a spatial modulation technique that balances the exploitation of global and local information. Additionally, we propose spatial domain augmentation to enhance generalization across varying spatial regions. Through ex- tensive experiments, we have demonstrated the superior performance of MARBLE in various PDE modeling tasks, including dynamics modeling and geometric prediction.
While MARBLE significantly enhances learning capacity, it's important to note that the complexity of GridMix may introduce additional computational overhead and memory requirements. A promis- ing avenue for future research would to explore efficient implementations of GridMix, potentially drawing inspiration from vector quantization and tensor factorization techniques commonly em- ployed in grid-based representations (Takikawa et al. 2022 Chen et al. 2022). We also observe that MARBLE performs below GEO-FNO the Pipe dataset, likely due to the limitations of the SIREN backbone, as noted by Serrano et al. (2023). While SIREN is effective for isotropic frequency dis- tributions, it may struggle with data featuring strong directional anisotropy due to its sensitivity to frequency-related byperparameters. Future work could explore enhanced INR designs (Cho et al. 2024] that better capture directional frequency variations.
10

---

Published as a conference paper at ICLR 2025
ACKNOWLEDGMENTS
The work is supported in part by the National Natural Science Foundation of China under Grant 62276150 and the Guoqiang Institute of Tsinghua University.
REPRODUCIBILITY STATEMENT
We provide detailed descriptions of dataset generation for the benchmarks Appendix A The experimental setups for our method outlined in Appendix B Additionally, our code is included in the supplementary materials.
REFERENCES
Matthias Bauer, Emilien Dupont, Andy Brock, Dan Rosenbaum, Jonathan Richard Schwarz, and Hyunjik Kim. Spatial functa: Scaling functa to imagenet classification and generation. arXiv preprint arXiv:2302.033130 2023.
Johannes Brandstetter, Daniel Worrall, and Max Welling. Message passing neural pde solvers. In International Conference Learning Representations, 2022.
Keaton J Burns, Geoffrey M Vasil, Jeffrey S Oishi, Daniel Lecoanet, and Benjamin P Brown. Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Re- view Research, 2(2):023068, 2020.
Anpei Chen, Zexiang Xu, Andreas Geiger, Jingyi Yu, and Hao Su. Tensorf: Tensorial radiance fields. In European conference on computer vision, pp. 333-350. Springer, 2022.
Anpei Chen, Zexiang Xu, Xinyue Wei, Siyu Tang, Hao Su, and Andreas Geiger. Factor fields: A unified framework for neural fields and beyond. arXiv preprint arXiv:230..01226, 2023.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances neural information processing systems, 31, 2018.
Junwoo Cho, Seungtae Nam, Hyunmo Yang, Scok Yun, Youngjoon Hong, and Eunbyung Park. Separable physice-informed neural networks. Advances in Neural Information Processing Sys- tems, 36, 2024.
Emilien Dupont, Hrushikesh Loya, Milad Alizadeh, Adam Golinski, Yee Whye Teh, and Arnaud Doucet. Coin++: Neural compression across modalities. Transactions on Machine Learning Research. 2022.
Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-leaming for fast adaptation of deep networks. In International conference on machine learning, pp. 1 1126-11333 PMLR, 2017.
Joseph Galewsky, Richard K Scott, and Lorenzo M Polvani. An initial-value problem for testing numerical models the global shallow-water equations. Tellus A: Dynamic Meteorology and Oceanography, 56(5):429 440, 2004.
Christian Grossmann. Numerical treatment of partial differential equations. Springer, 2007.
Kenneth Huebner, Donald Dewhirst, Douglas E Smith, and Ted G Byrom. The finite element method for engineers. John Wiley & Sons, 2001.
Chiheon Kim, Doyup Lee, Saehoon Kim, Minsu Cho, and Wook-Shin Han. Generalizable implicit neural representations via instance pattern composers. In Proceedings of the IEEF/CVF Confer- on Computer Vision and Pattern Recognition, pp. 11808 11817, 2023.
David M Knigge, David R Wessels, Riccardo Valperga, Samuele Papa, Jan-Jakob Sonke. Efstratios Gavves, and Erik Bekkers. Space-time continuous pde forecasting using equivariant neural fields. arXiv preprint arXiv:2406.06666 2024.
Dmitrii Kochkov, Jamie A Smith, Ayya Alieva, Qing Wang, Michael P Brenner, and Stephan beamiing-ccceeraater Hoyer. Machine learning accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences, 118(1)):210178811 18, 2021.

---

Published as a conference paper at ICLR 2025
Doyup Lee, Chiheon Kim, Minsu Cho, and WOOK SHIN HAN. Locality-awar generalizable implicit neural representation. Advances in Neural Information Processing Systems, 36, 2024a.
Joo Chan Lee, Daniel Rho, Seungtae Nam, Jong Hwan Ko, and Eunbyung Park. Coordinate-and aware modulation for neural fields. In The Twelfth International Conference on Learning Representa- tions, 2024b.
Zongyi Li. Nikola Borislavov Kovachki. Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar, et al. Fourier neural operator for parametric partial differential equa- tions. In International Conference on Learning Representations, 2021.
Zongyi Li, Daniel Zhengyu Huang. Burigede Liu, and Anima Anandkumar. Fourier neural oper- ator with learned deformations for pdes on general geometries. Journal of Machine Learning Research, 24(388): 1-26, 2023a.
Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural oper- ator with learned deformations for pdes on general geometries. Journal of Machine Learning Research, 24(388):1 2023b.
Lingjie Liu, Jiatao Gu, Kyaw Zaw Lin, Tat-Seng Chua, and Christian Theobalt. Neural sparse voxel fields. Advances in Neural Information Processing Systems, 13:15651-15663, 2020.
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, (15):218-2299 2021.
Lu Lu, Raphaël Pestourie, Steven G Johnson, and Giuseppe Romano. Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Physical Review Research, 4(2):023210, 2022.
Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1)999-106, 2021.
Roberto Molinaro, Yunan Yang. Björn Engquist, and Siddhartha Mishra. Neural inverse operators for solving pde inverse problems. In International Conference on Machine Learning, pp. 25105- 25139. PMLR, 2023.
Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. Instant neural graphics prim- itives with a multiresolution hash encoding. ACM transactions on graphics (TOG), 41(4):1-15, 2022.
Vien Minh Nguyen- Thanh, Xiaoying Zhuang, and Timon Rabezuk. A deep energy method for finite deformation hyperelasticity. European Journal of Mechamics-A/Solids, 80:103874, 2020.
Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 165-174, 2019.
Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. Film: Visual reasoning with a general conditioning layer. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018
Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks, In International conference on machine learning, pp. 5301-5310 PMLR, 2019.
Maziar Raissi, Paris Perdikaris, and George Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686-707, 2019.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomed- ical image segmentation. In Medical image computing and cosputer-assisted intervention- MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceed- ings, part III 18, pp. 241, Springer, 2015.
12

---

Published as a conference paper at ICLR 2025
Esteban Samaniego, Cosmin Anitescu, Somdatta Goswami, Vien Minh Nguyen-Thanh, Hongwei Guo, Khader Hamdia, Xiaoying Zhuang, and Timon Rabezuk. An energy approach to the solu- tion of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362: 1 12790, 2020.
Louis Serrano, Lise Le Boudec, Armand Kassaī Koupaï, Thomas X Wang. Yuan Yin, Jean-Noël Vittaut, and Patrick Gallinari. Operator learning with neural fields: Tackling pdes on general geometries. Advances in Neural Information Processing Systems, 36, 2023.
Jie Shen, Tao Tang, and Li-Lian Wang- Spectral methods: algorithms, analysis and applications, volume 41. Springer Science & Business Media, 2011.
Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Advances in neural information processing systems, 33:7462-7473, 2020.
Zhiqing Sun, Yiming Yang. and Shinjae Yoo. A neural pde solver with temporal stencil modeling. In International Conference on Machine Learning, pp.
PMLR, 2023.
Towaki Takikawa, Joey Litalien, Kangxue Yin, Karsten Kreis, Charles Loop, Derek Nowrouzezahraii Alec Jacobson, Morgan McGuire, and Sanja Fidler. Neural geometric level of detail: Real-time rendering with implicit 3d shapes. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1135--1167, 2021.
Towaki Takikawa, Alex Evans, Jonathan Tremblay, Thomas Müller, Morgan McGuire, Alec Ja- cobson, and Sanja Fidler. Variable bitrate neural fields. ACM SIGGRAPH 2022 Conference Proceedings, pp. 1 1-9, 2022.
Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keii, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural information processing systems, 33:7537-7547, 2020.
Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural
operators, In The Eleventh International Conference on Learning Representations, 2023.
Honghui Wang, Lu Lu, Shiji Song, and Gao Huang. Learning specialized activation functions for physics-informed neural networks. arXiv preprint arXiv:2308.04073, 2023.
Honghui Wang, Yifan Pu, Shiji Song, and Gao Huang. Advancing generalization in pinns through latent-space representations. arXiv preprint 2rXiv2241..19125,1 2024.
David R Wessels, David M Knigge, Samuele Papa, Riccardo Valperga, Sharvaree Vadgama, Efstra- tios Gavves, and Brik Bekkers. Grounding continuous representations geometry: Equivariant neural fields. arXiv preprint arXiv:2406.05753, 2024.
Yuan Yin, Matthieu Kirchmeyer, Jean- Yves Franceschi, Alain Rakotomamonjy, et al. Continuous pde dynamics forecasting with implicit neural representations. In The Eleventh International Conference on Learning Representations, 2022.
13

---

Published as a conference paper at ICLR 2025
APPENDIX
A BENCHMARKS
A.1 DYNAMIC'S MODELING
2D Si-NNaier--totee (Navier-Stoken) We examine the 2D Navier-Stokes equation as detailed in Li et al. (202T): Yin et al. (2022); ible fluid within rectangular domain Q - 1, The governing PDE is given by:
( (Serrano et al. (023). which simulates the behavior of an incompress-
aw(a, 1)
u(x, 1) V2, 1) + (w (x, 1) f,x E (-,, (0,7]
(11)
Öt
w(x 1) V ul(, () [ 1, (0,
(12)
Vu(z, 0, |-1, 1 [0.
(13)
where u denotes the velocity field, w represents the vorticity, v is the fluid viscosity, and f is the forcing term defined as:
f(11, x22) = 0.1 (sin(2 + x2)) + cos(2 + x2))) Vx S
(14)
The system is subject to periodic boundary conditions. We generated data by sampling initial condi- tions in Li et al. (202T), producing various trajectories on 256 x 256 spatial grid with time step 5t = 1. To capture significant dynamics, we retain the trajectory starting from the 20th timestep. Each resulting trajectory contains 40 snapshots. We split these trajectories into two segments: the first 20 frames are utilized for training (denoted as In-f), while the remaining 20 frames are reserved for testing the model's extrapolation capability (referred to as Out-t). In total, we generated 256 trajectories for training and 16 for evaluation.
3D-Spherical Shallow-Wate (Shallow- Shalloo-WWteer). We investigate the shallow-water equation on a spherical surface, which models atmospheric dynamics on Earth:
du dl f k: x W vAu
(15)
dh
dt AV + vAh
(16)
Here, is a unit vector perpendicular to the sphere's surface, sphere, which can be converted to vorticity tion was carried out using the Dedalus framework (Burns et al. 2020), following the configuration described in Yin et al. (2022), where symmetric behavior is observed in both hemispheres. The ini- tial velocity uo consists of two symmetric bands, one in each hemisphere, aligned with the latitude circles. For each point at latitude and longitude 0 E [- - #1
the velocity field tangent to the
V x u, and h is the sphere's height. Data genera-
[-
Mmax 6n
exp
(6 (o-do)le-o) 01)
if (du.01)
uo( 0)
exp
(0+t0)
if E (-#1, - oo). otherwise.
(17)
where Umar the peak velocity, Φo のi = water height ho is computed by solving a boundary value problem as described in Galewsky et al. 2004). with perturbation added by h/:
Φn. and 6n exp( (01 - do
The initial
1/6 (1, 0) = cos exp (- (-(2))
02 (-(02
+ exp (- Φ2 + + )
0
exp
(18)
B
where Φ2 = 3. = 120 m. 3. and 3 = 1 following Galewsky et al. (2004). Simulations were executed on snituue--lngitudee grid using Dedalus (Burns et at. 2020). starting with an initial grid size of 128 x 256, which was downsampled to 64 x 128. Data generation involved sampling umaz from a uniform distribution (60, 80), with snapshots taken every hour over a period of 320 hours, yielding 320 timestamps per trajectory. We created 16 trajectories for training and 2 for testing. However, since the early snapshots exhibited less dynamic activity, only the last 160 snapshots were retained. These long trajectories were then segmented into sub-traject trojeeeectooies of 40 timestamps each, resulting in 64 training trajectories and testing trajectories. Finally, the data was rescaled: height h was scaled by 3 x 103, and vorticity we scaled by factor of 2.
15
14

---

Published as a conference paper at ICLR 2025
A.2 GEOMETRIC-AWARE INFERENCE
We utilize datasets from Li et al. 2023b) and use the original authors' train/test split.
Euler's Equation (Naca-Euler).
This experiment focuses on transonic flow over an airfoil, gov-
erned by the Fuler equations:
Opg V (ppu) 0, apyu v pp) OE +V- (E++))u)
(19)
where Ps denotes the fluid density, represents the velocity vector, p is the pressure, and E is the total energy. Viscous effects are disregarded. The boundary conditions are set as follows: Pso = 1, Poo 1.0, M& 0.8, and АoA O, where Mx is the Mach number and АoA stands for the angle of attack. no-penetration condition is applied at the airfoil surface. The airfoil shape is parameterized using the design element method. Specifically, the initial NACA-0012 shape is mapped onto a "cubic" design element with control nodes, and the initial shape is morphed to different one following the control nodes' displacement field. These control nodes can only move vertically, with displacements following a uniform distribution d ~ 4[-0.05, 0.05]. The dataset consists of 1000 training examples and 200 test examples, generated using a second-orde implicit finite volume solver. The mesh point locations and the Mach number at these points serve as the input and output data, respectively.
Hyper-elastic Material (Elasticity).
We consider the governing equation of a solid body as:
8²u Ps 0t
0
where P. the mass density, represents the displacement vector, and is the stress tensor. To close the system, constitutive model links the strain tensor E to the stress tensor. Our study investigates a unit cell problem Ω = [0, 1] x [0. 1] with void at the center. The void's radius follows r 0.2 + I+)xp(F) with F ~ (0,, 42 - V + 32)-1). The bottom edge of the unit cell is clamped, and tension traction & [0, 100) is applied on the top edge. We use the Sewlin-Saunders material, with energy density parameters C1 = 1.863 x 10" and C2 = 9.79 10?. Data was generated using finite element solver with around 100 quadratic quadrilateral elements. The input data, represented as point clouds containing roughly 1000 points, was used to predict stress as the target output.
0.2
Navier-Stokes Equation (Pipe). This scenario involves simulating incompressible flow through pipe, governed by the Navier-Stokes equations:
Ov
+ (v. at
V)u - 2V2
v 0
where e denotes the velocity vector, p is the pressure, and 0.005 represents the viscosity. The inlet imposes a parabolic velocity profile with a maximum velocity of o = [1,0]. A free boundary condition is applied at the outlet, and no-slip conditions enforeed the pipe's appes surface. The pipe has a length 10 and a width of Its centerline is parameterized by 4 piecewise cubic polynomials, controlled by the vertical positions and slopes spatially uniform nodes. The vertical positions at these control nodes follow a uniform distribution ~ [([-2, 2]. while the slopes adhere to d ~ 2[[1, The dataset includes 1000 training examples and 200 test examples, generated using an implicit finite element solver with approximately 4000 Taylor-Hood Q2-Q1 mixed elements. The input data comprises the mesh point locations (129 129), and the horizontal velocity these points is the output.
B EXPERIMENTS SETUPS
To train model, we follow the experimental setup and hyperparameter choices outlined in the original CORAL (Serrano et al., 2023) paper. The additional hyperparameters for MARBLE are listed in Table
15

---

Published as a conference paper at ICLR 2025
Table The choices hyperparameters for GridMix and spatial domain augmentation (SDA)
Plyper-prraeete
Navier-Stokes
Shallow- Water
NACA-Euler
Elasticity
Pipe
grid resolution
8x8
8x16
32x8
8x8
8x8 8
basis functions count
32
32
64
64
64
GridMix
latent dimension
32
256
128
128
128
basis learning rate
le-2
le-2
le-2
le-2
le-2
SDA sampling ratio
0.4
0.4
C ADDITIONAL VISUALIZATIONS
We present visualizations of the predictions made by MARBLE on the Navier-Stokes dataset in Figure 3 on the Elasticity dataset in Figure 6
Jn
Out
Reference
CORAL
Ours
Figure 5: Visualization of predictions of MARBLE Navier-Stokes dataset with 20%.
Reference
CORAL
Ours
Reference
CORAL
Ours
Absolute Error
Absolute
Frror
Figure 6: Visualization of predictions of MARBLE on Elasticity dataset.
D EXTENDED RELATED WORK
Classical Grid-based Methods, such as finite difference (FDM) (Grossmann 2007). finite el- ement (FEM) (Huebner et al.] 2001), and spectral methods (Shen et al. 2011). are foundational techniques for solving partial differential equations. These approaches discretize the spatial domain into structured unstructured grid, allowing the numerical approximation of derivatives and the solution of governing equations. FDM rely on regular grids and approximate derivatives through dif- ferences between neighhoring grid points. While simple and efficient for structured domains, they
16

---

Published as a conference paper at ICLR 2025
struggle with complex geometries adaptivity to dynamic domain changes. FEM unstructured grids and local basis functions to discretize the domain, providing higher flexibility for complex ge- ometries and localized refinement. However, their computational cost can increase significantly with mesh refinement. Spectral Methods decompose the solution into global basis functions (e.g., Fourier Chebyshev polynomials) and excel in smooth domains due to their high accuracy. Yet, they face challenges in handling discontinuities or irregular domains, Despite their robustness and accuracy, these methods are typically tailored fixed domains, making generalization to unseen configura- tions or dynamic problems challenging. Furthermore, their computational complexity scales poorly with increasing grid resolution, especially for sigh-dimensional problems.
Spatial Modulation for Neural Fields. Grid-based representations (uülle et al. 2022 have
driven significant progress in reconstructing single continuous signals with neural fields. This foun- dation has been extended to encode multiple data instances through spatial modulation (Bauer et al.
2023 Kim al. 2023 Lee al.] 2024a Wessels al. 2024). For instance, [Bauer al. (023)
introduced a scalable approach for representing high-reso lution images using spatial modulation. Subsequent works, such Kim al. (2023) Lee al. (2024a), further enhanced the expressive power of this technique by incorporating Transformer based encoders. WWssels et al. (2024) took this step further by embedding geometric information into modulation spaces, enabling steerabil- ity properties. While the majority of research in this area has focused on computer vision tasks, recent work (Knigge et al., 2024) has explored grounding geometric information within modulation spaces to preserve PDE symmetries, thereby improving generalization and data efficiency for PDE modeling. Our work distinguishes itself by uniquely focusing on enhancing domain generalization while preserving the locality of spatial modulations-ai underexplored aspect that is critical for PDE modeling.
E EXTENDED COMPARISONS
We compare our method with Factorized-FNO (Tran al. 2023) in the dynamics modeling set- ting, as shown in Table 7 Specifically, we use a 12-layer Factorized- with 16 modes and a width of 64. For the irregular grid setting, linear interpolation is used to preprocess the data for compatibility with Factorized-FNi
As observed, Factorized- demonstrates improved perfor- mance compared to FNO across all settings. However, method, MARBLE, consistently achieves superior performance.
Table 7: Dynamics Modeling Test results. Metries in MSE. The best results are shown in bold, while the becond-best results are underlined. The "lin. int." abbreviates linear interpolation of irregular grid data onto regular grid.
Ni. 1. Aic
datasct
Navier-Stokes
In-t
Out-r
DeepONet
4.72e-2 2.84
9.58e 1.83e
FNO
5.68e-4 7.62e-5
8.95e.3 + 1.50e
3.18e
1.12e.3
100% rogular grid
MP-PDE
4.39c-4 1: 8.78e-5
4.46e-3 +: 1.28e-3
DINo
1,270-3 2.22o-5
1.le-2 : 2.28c-3
CORAL
1.86-4 1.44-55
1.02e-3
+ 8.62e-5
MARBLE (Ours)
3.52e-5
5.Me-6
5.04e-4 3.68e.5
DeepONet
8.37e-1 1. 2.07e-2
7.80e 1. 2.36e
FNO lin.
8.03e-4
Factorized-FNO- lin. int.
3.42c-3 1.15e4
6.68c-3 + 2.87c-4
20%
MP-PDE
3,98--2 1,69e-2
1,31e-1 + 5.34e-2
imegular grid
DINo
9.99 1..44.6..77 6.71e.3
8.27e-3 1 5.6le:3
CORAL
2.18e:3 6.88c-4
6.67e:3 2.0ee33
MARBLE (Ours)
1.62-4 2.42e-5
9.27e- 1.44::1.444 1.44e-4
DeepONet
7,86e 5.48e
7.48e 1. 2.76e:2
ENO lin. int.
3,87e 1.4e-2
5.19e-2 :: 1.100-2
Factorized-FNO lin. int.
3.1 3.06e-3
4.88c=2 # 1,58c-3
5% imegular grid
1.92-11 9.276-2
4.7ce1
2.17e-1
DIND
8.65e:2
1.16e
9.36e 2 1 9.34e
2.44e22
1.96e-2
1.78e:2
MARBLE (Ours)
1.4e-3 - 5.66e-4
4.73e-4 + 1.33e-3
17

---

Published as a conference paper at ICLR 2025
F COMPUTATIONAL COMPLEXITY
In this section, we provide an analysis the computational complexity of our method. Specifically, we unroll trajectory up 40 with batch size of I and report runtime and memory usage in All experiments were conducted on an NVIDIA A100 GPU.
Figure 7
Time Complexity. Our method shows a moderate increase in runtime compared to CORAL. For example, at resolution of 128 x 128, our approach is approximately 1.6x slower than CORAL This additional overhead arises from steps like spatial modulation and interpolation in our frame- work. Nonetheless, ical methods. The pseudo-spectral method, the numerical baseline used to generate the dataset (Li et al. 2021p. requires approximately order of magnitude more time than method at the same resolution.
method still maintains substantial efficiency compared to traditional numer-
Memory Usage. Our method incurs about 6.5x the memory overhead compared to CORAL across varying resolutions, primarily due to additional parameters and intermediate computations required by our enhanced representation. However, the memory cost scales predictably and remains well within the capabilities of modern GPUs for the tested resolutions, ensuring feasibility for larger datasets and higher resolutions in practical applications. As discussed in Section | this memory overhead could potentially be mitigated through techniques such as vector quantization (Takikawal ict al. 2022) and tensor factorization (Chen et al. 2022). offering a promising direction for future rescarch.
Time (s)
Memory (GB)
109
CORAL
CORAL
10a
Ours
Pseudo-spectral
10-1
10-1
32
64
128
32
64
128
Reselution
Resolution
Figure 7: Computational complexity of MARBLE.
G VISUALIZATION OF LATENT EMBEDDINGS
To analyze the impact of increasing the number of basis functions on model performance, we vi- sualize the latent embeddings generated with 32 and 64 basis functions in Figure g Specifically, we randomly select 5 dimensions from the latent embedding = and plot their changes over time. As observed, the trajectories generated with 64 basis functions exhibit greater fluctuations compared to those with 32 basis functions. This increased variability may complicate the learning process for Neural ODEs, potentially leading decrease in forecasting performance.
33 beih
6.8
32
32 soth
8.6
0.0
-0.35
-0.00
-3.55
-03
0.20
-12
1.25
-32
Figure 8: Latent embeddings generated with 32 and 64 basis functions.
18


Reviews:
[{"review_id": "CvHJpzzQ1x", "reviewer": "Reviewer_bgUL", "summary": "The paper proposes an new architecture called GridMix, which enhances implicit neural fields (INRs) for solving PDEs. Building on multi-resolution INR advances from visual computing ([1], [2]), the authors adapt these methods to improve PDE modeling. More specifically, they solve the problem using CORAL[3] approach, where each of the INR is encoded using a multi-resolution feature grid (for spatial modulation), combined with a new spatial domain augmentation strategy to boost generalization across spatial domains. Experiments demonstrate GridMix’s efficacy in dynamic modeling tasks (temporal extrapolation and spatial interpolation) and in steady-state predictions based on domain geometry.", "questions": "– In Table 1, why single resolution feature grid (Single-channel GridMix) gives worse results\nthan Global Modulation? As the spatial feature representation should give already good results\ngiven the initial motivation\n– I dont fully understand the explanation of why more basis functions lead to worse results at\nsome point? As in case there are too many basis, they can be zeroed out with corresponding\nweights\n- Figure 3 is hard to read – neet legend on the Y axis. It is not clear to me from the footnote what’s going\non.\n\n\nReferences:\n[1] “Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes” by Towaki\nTakikawa et. al.\n[2] “Instant Neural Graphics Primitives with a Multiresolution Hash Encoding” by Thomas Muller\net. al.\n[3] “Operator Learning with Neural Fields: Tackling PDEs on General Geometries” by Louis\nSerran et. al.\n[4] “Factor Fields: A Unified Framework for Neural Fields and Beyond” by Anpei Chen et. al.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "– Adaptation of multi-feature grids for auto-decoding: the paper presents a novel approach for leveraging multi-feature grids in the auto-decoding setting, effectively adapting INRs to capture complex spatial variations while maintaining efficiency.\n– Extensive comparisons and ablations study\n– Modeling spatial modulation using basis functions (and their coefficients) looks promising", "weaknesses": "– Some quantitative results, especially those comparing single-resolution grids with global modulation in Table 1, appear counterintuitive. The single-channel GridMix approach yields lower performance in some cases compared to simpler global modulation, which contradicts the initial motivation that spatial features should perform better.\n– Missing mentioning Factor Fields [4] paper that also works in similar framework and solves similar problems but not in PDE domain", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uxqenGkIP3", "reviewer": "Reviewer_M2jE", "summary": "To address the limitations of global modulation, this paper proposes GridMix technology, a novel spatial modulation method. GridMix combines the advantages of grid-based representation and spatial modulation by regularizing the modulation space into a set of shared basis functions, thereby enhancing the model's ability to learn global structures. Additionally, a spatial domain enhancement strategy is introduced to simulate domain changes that may occur during inference, further improving the model's generalization capability. Experiments demonstrate that GridMix excels in dynamic modeling and geometric prediction tasks, particularly in cases of temporal extrapolation and spatial subsampling, outperforming existing models.", "questions": "1. What are the basic settings of the grid? The current article only mentions grid density, but it lacks details on discretization methods, grid shapes, and adaptive methods (how to handle complex geometries). Is the discretization method used by the authors consistent with those in numerical simulations?\n2. What is the computational efficiency of the proposed method? There is a need to compare the proposed method with numerical simulations and other data-driven PDE modeling methods.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. Innovation: The introduction of GridMix overcomes the shortcomings of traditional global modulation in capturing local features by incorporating grid-based representation.\n2. Experimental Validation: The effectiveness and versatility of MARBLE are validated through extensive experiments, including dynamic modeling and geometric prediction. The experimental design is robust, covering various testing scenarios and enhancing the credibility of the results.\n3. Algorithm Flexibility and Universality: The MARBLE framework demonstrates applicability across multiple tasks by combining GridMix and spatial domain enhancement, indicating its broad potential for application in PDE modeling.", "weaknesses": "1. Insufficient Literature Review: There is a lack of introduction to grid-based methods, such as discretization techniques and computational efficiency.\n2. Complexity of Proposed Method: Although the proposed new method is innovative, its complexity may lead to challenges in implementation, particularly regarding parameter tuning and optimization in different application scenarios. It is unclear whether this method retains a speed advantage compared to numerical simulations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "o1cp6unSUs", "reviewer": "Reviewer_oXC3", "summary": "Summary: The paper introduces MARBLE, a new training framework for INR-based neural solvers for PDEs. MARBLE has two components: 1. GridMix and 2. Spatial domain augmentation (SDA). The paper builds off of CORAL [1], primarily changing the shift modulation. \n\nGridMix replaces $\\phi_{a} = h_{a}(z_{a_i})$ with a linear combination of learned basis functions. As such, $\\phi$ takes the form $\\phi_{a}(x) = \\Sigma_{m=1}^M c_i^m(z_a)\\Phi^m_i(x)$. The set of coefficients $c^m$ are a hypernetwork function of the latent embedding $z_a$ and $\\Phi$ functions are learned along with the INR weights. Points on the domain $x$, are binned into patches of $H\\times W$, and $\\Phi$ is a tensor of shape $H\\times W \\times C$, representing channel-wise modulation binned according to $H \\times W$.\n\nSDA is a form of data augmentation which stochastically drops points from the input domain.\n\n*Experiments*: The paper conducts two sets of experiments: one on learning dynamics (NS, Shallow water) and the second on problems mapping from reference grid to steady states (Euler, NS Pipe, Hyper-elastic material). Ablations are performed over NS.", "questions": "- Why are the models used for the two evaluation settings different? While FNO is a good neural operator baseline, there have been many advancements (e.g., FFNO [2]) which could be run on the dynamics modeling case. *Requested Exp*: Use a better FNO variant for a more apt comparison to spectral neural operators (NOs). Can the authors justify the choice of baselines for each evaluation setting and why certain models were not used in both contexts? \n- Table 2: What does “FNO + lin. int.” mean? Additionally, how is an irregular grid being used with FNO? Please add a brief explanation of these terms and methods to either the table caption or main text. A spectral NO designed for irregular meshes would be a more apt baseline (e.g., Geo-FNO [3]). *Requested Exp*: Geo-FNO / FNO variant designed for irregular meshes.\n- Table 2: No explanation for the N.A. values.\n- “Baseline results for comparison are sourced from Serrano et al. [1]”. What does this mean? Please provide a detailed explanation of how the baseline results were obtained, including any steps to ensure consistency with the original datasets if new data was generated (seems to be indicated in Appendix A.1).\n- There are a few cases where the proposed method performs marginally worse than some of the baselines (Shallow water $\\pi=5\\%$, Pipe). What about these cases makes MARBLE perform worse? It would benefit the paper to include a brief analysis of these specific cases in the discussion section, exploring possible reasons for the underperformance and how future work might add it.\n- Dynamics modeling (architecture details): A single modulated INR is used in this case. This is different from Fig 1 and the setup where two modulated INRs are used: $f_{\\theta_{a}}$ and $f_{\\theta_{u}}$. What is the explicit differences from the general setup in the method section? Please clarify how the input and output functions are handled in this experimental setting.\n\n[1] Serrano, L., Le Boudec, L., Koupaï, A., Wang, T., Yin, Y., Vittaut, J.N., & Gallinari, P. (2024). Operator learning with neural fields: tackling PDEs on general geometries. In Proceedings of the 37th International Conference on Neural Information Processing Systems. Curran Associates Inc..\n\n[2] Alasdair Tran, Alexander Mathews, Lexing Xie, & Cheng Soon Ong (2023). Factorized Fourier Neural Operators. In The Eleventh International Conference on Learning Representations .\n\n[3] Li, Z., Huang, D., Liu, B., & Anandkumar, A. (2024). Fourier neural operator with learned deformations for PDEs on general geometries. J. Mach. Learn. Res., 24(1).", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The problem is of strong interest and the proposed solution is novel.\n- Multiple complex datasets are used for experimental results, representing a variety of tasks. The proposed method has substantive improvements over the baselines.\n- The ablations indicate a clear improvement to CORAL on both the in and out of distribution time-horizons.", "weaknesses": "High level weaknesses, articulated further under questions.\n- Some of the mathematical notation and setup for GridMix is confusing. \n    - The computation of the position-dependent shift modulation scalar is not clear. The text says: “we derive the position-dependent shift modulation scalar $\\phi_i(x)$ at location x through interpolation between neighboring features.” From fig 2, it seems that position x is mapped to a bin which then selects the corresponding modulation value. Please provide a step-by-step explanation as to how x is mapped to the modulation variable, including the details of how the interpolation is performed.\n    - Equation 7 is presents $\\phi_i(x)$ as independent of $z_{a}$. Equation 7 also omits the hypernetwork, obfuscating the dependence on $z_{a}$. Please clarify in the text that this dependence by providing a supplemental equation that illustrates this relationship or including the hypernetwork in equation 7.\n- The baseline models are not consistent between the two classes of experiments.\n- Some experimental details are not clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oKpNVlqAZL", "reviewer": "Reviewer_pmNQ", "summary": "In recent years, neural field-based methods for dynamics modelling have seen increasing interesting due to their flexible nature w.r.t. geometry and sparsity of the input. Most methods that have previously been proposed employ conditional neural fields, where a base model is conditioned by a global latent that is shared over all spatial positions in the input function. Although this has shown promising results in complex forecasting settings, the use of global modulation limits the ability of the NeF to reconstruct fine details. As such, authors propose instead learning a spatial basis of modulations in which a sample can be expressed through a combination of coefficients. This allows for location-specific modulation while retaining the globally informed continuous properties of global modulations. Authors show results for a range of experiments, assessing temporal generalization and robustness to subsampling, in which their method consistently improves over baseline architectures.", "questions": "- Is there a particular reason you chose to optimize the decoders and neural ode models separately? Would it not be possible to optimize these end-to-end?\n- Could you provide details on the computational complexity of your method compared with CORAL? How much overhead is there in terms of memory and time complexity compared to this baseline?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The method is effective and simple, instead of learning a global modulation which limits fine-grained reconstruction or a purely local latent which limits generalizability, a spatial modulation is expressed in a learned basis. The experimental results show good performance. This approach is general and could be applied to a wide range of other interesting forecasting tasks.\n- The paper is well-written and builds towards the method incrementally. The method is very well motivated, also supported by empirical evidence.\n- The paper contains illustrative figures that aid in understanding the work.", "weaknesses": "- The only limitation I see, although I do not deem it a major one, is the relatively limited novelty of the method. There have been quite a number of previous works exploring spatial modulation for NeF-based reconstruction and downstream performance, notably [1, 2, 3]. For instance, in computer vision NeF literature a widely carried modulation/embedding is hash encoding [4], which your approach to modulation learning seems somewhat related to. Within PDE solving literature, a concurrent work [5] proposes a NeF-based solving framework based on DINo which also learns a spatial latent using MAML, though their approach is more tailored towards PDEs with known symmetries. It might be good to mention these works and compare (maybe in an extended related work section in the appx.), as the (vision) NeF community and the Neural Operator (physics informed learning) communities have been working somewhat in parallel.\n- No details are given regarding the computational complexity of the method. Would your method scale easily to higher resolution / larger data? I think this is an important consideration, and as such it would be good to provide details on computational complexity and memory efficiency.\n\nOther than this, I think this work is interesting and shows valuable results for the deep-learning based PDE solving community. I think this paper would be a valuable contribution to the conference.\n\n[1] Bauer, M., Dupont, E., Brock, A., Rosenbaum, D., Schwarz, J. R., & Kim, H. (2023). Spatial functa: Scaling functa to imagenet classification and generation. arXiv preprint arXiv:2302.03130.\n[2] Lee, D., Kim, C., Cho, M., & HAN, W. S. (2024). Locality-aware generalizable implicit neural representation. Advances in Neural Information Processing Systems, 36.\n[3] Wessels, D. R., Knigge, D. M., Papa, S., Valperga, R., Vadgama, S., Gavves, E., & Bekkers, E. J. (2024). Grounding Continuous Representations in Geometry: Equivariant Neural Fields. arXiv preprint arXiv:2406.05753.\n[4] Müller, T., Evans, A., Schied, C., & Keller, A. (2022). Instant neural graphics primitives with a multiresolution hash encoding. ACM transactions on graphics (TOG), 41(4), 1-15.\n[5] Knigge, D. M., Wessels, D. R., Valperga, R., Papa, S., Sonke, J. J., Gavves, E., & Bekkers, E. J. (2024). Space-Time Continuous PDE Forecasting using Equivariant Neural Fields. arXiv preprint arXiv:2406.06660.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)