Breaking the Discretization Barrier of Continuous Physics Simulation Learning
Authors: Fan Xu
Hao Wu
Nan Wang
Lilan Peng
Kun Wang
Wei Gong
Xibin Zhao
Venue: NeurIPS 2025 poster ()
Keywords: dynamical system modeling
Primary area: machine_learning_for_sciences

Abstract: The modeling of complicated time-evolving physical dynamics from partial observations is a long-standing challenge. Particularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to capture highly nonlinear features in a variety of scientific and engineering problems. However, existing data-driven approaches are often constrained by fixed spatial and temporal discretization. While some researchers attempt to achieve spatio-temporal continuity by designing novel strategies, they either overly rely on traditional numerical methods or fail to truly overcome the limitations imposed by discretization. To address these, we propose CoPS, a purely data-driven methods, to effectively model continuous physics simulation from partial observations. Specifically, we employ multiplicative filter network to fuse and encode spatial information with the corresponding observations. Then we customize geometric grids and use message-passing mechanism to map features from original spatial domain to the customized grids. Subsequently, CoPS models continuous-time dynamics by designing multi-scale graph ODEs, while introducing a Markov-based neural auto-correction module to assist and constrain the continuous extrapolations. Comprehensive experiments demonstrate that CoPS advances the state-of-the-art methods in space-time continuous modeling across various scenarios. The source code is available at~\url{https://github.com/Sunxkissed/CoPS}.

Breaking the Discretization Barrier of Continuous Physics Simulation Learning
Fan Xu' Hao Wu"*, Nan Wang"1, Lilan Peng", Kun Wang Wei Gong",, Xibin Zhao
University of Science and Technology of China TTsnnguua University
Beijing Jiaotong University "Southwest Jiaotong University NNayyang Technological University
Equal Contribution, Conreponning author
Main Contact: markxu@mail. messt..edu.co
Abstract
The modeling of complicated time-evolving physical dynamics from partial ob- servations is long-standing challenge. Partieularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to capture highly nonlinear features in variety of scientific and engineering prob- lems. However, existing data-driven approaches are often constrained by fixed spatial and temporal discretization. While some researchers attempt achieve spatio-temporal continuity by designing novel strategies, they either overly rely on traditional numerical methods or fail truly overcome the limitations imposed by discretization. To address these, we propose CoPS, purely data-drives meth- ods, effectively model continuous physics simulation from partial observations. Specifically, we employ multiplicative filter network to fuse and encode spatial information with the corresponding observations. Then we customize geometric mechanism to map features from original spatial
grids and use message-passingg domain to the customized grids. Subsequently, CoPS models continuous-time dynamics by designing multi-scale graph ODEs, while introducing Markov-bass based neural auto-correction module to assist and constrain the continuous extrapolations. Comprehensive experiments demonstrate that CoPS advances the state-of-the-a methods in space-time continuous modeling across various scenarios. The source code is available at https: //t///gitttt //github... cwm/ /ttss/nniisssd/oPs....
1 Introduction
Achieving accurate global modeling and forecasting of complex time-evolving physical system from a limited number of observations has been a long-standing challenge 52, 11]. This has widespread applications in chaotic physical systems such geophysics [12, 38], atmospheric science [9, 40], and fluid dynamics [56, 29, 55, 4], et al. For instance, in metcorology and occanography, sensors are often deployed in areas with scarce non-existent network connectivity. This renders the analysis and modeling of the system a formidable obstacle. From empirical knowledge, dynamical systems can be fully described by complicated and not yet fully elucidated partial differential equations (PDEs) [16]. Traditional numerical methods such as finite element [13] and finite volume [2] methods excessively rely prior knowledge of essential PDEs and suffer from high time complexity, making it difficult to accommodate the increasing demands for grid granularity. Recently, data-driven approaches partially overcome these by integrating advanced neural networks directly learn dynamic latent features with high nonlinearity from existing observations or simulations.
Although data-driven methods [18, 31, 45] have the capacity learn complex relationships from observations, they still have serious limitations in changeable actual scenes. One significant challenge in modeling physical fields from partial observations is that the field may not adhere to Cartesian
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

grid [34, I], and convolutional neural networks are inherently not adapted to such structures. When employing graph neural networks [22, 51], the graph topology is often determined by the locations of existing observation points, making the model difficult to generalize to unseen spatial locations. Additionally, some methods use padding or interpolation [6] to enforce spatiotemporal continuity. This may distort the inherent inductive bias of the observed data and significantly increases the computational burden sparse data scenarios.
Recently, to better explore the space- time continuous formulation from par- tial observations [19, 53]. few no- table works stand out. Example of continuous time and space dynamic system evolution is illustrated Fig- ure 1. For cpace-continuous learn- ing, the multiplicative filter network (MFN) [10], combined with implicit neural representations (INRs) [17], has been proposed to incorporate co- ordinate information through linear
Grid and timestep-free cvolution
Sparse observations
Free timestep
F0
T1
T2
T3
T
Figure Example of continuous time and space dynamic system evolution. (Non-discretized)
Fourier wavelet functions in efficient manner. However, this approach requires iterating for hundreds of steps to obtain the corresponding implicit neural representation for new samples, severely lacking real-time applicability. Further, for continuous-ting learning, neural ordinary differ- ential equations (Neural ODEs) [7] address the limitation of fixed time extrapolation windows by learning continuous ODEs from discrete data using explicit solvers such the Euler or Kunge-Kutta methods [3, 33]. However, the system's 's nonlinearity is high, Neural ODE may struggle to capture the essential features of the complex dynamics. Additionally, Neural ODE is solved through numer- ical integration, which leads to errors accumulating as time progresses. As a result, conventional Neural ODEs may lead poor stability in long-term predictions and potentially worse fitting.
To address these, we propose CoPS, purely data-driven methods, to effectively achieve space- time continuous prediction of dynamical systems based partial observations. Specifically, employ multiplicative filter networks to fuse and encode spatial information with the corresponding observations. Then customize geometric grids and use message passing mechanism map features from original spatial domain to the customized grids. Subsequently, CoPS models continuous dynamics by designing multi-scale Graph ODEs, while introducing local refinement feature extractor to assist and constrain the parametric evolutions. Finally, the predicted latent features are mapped back original spatial domain through GNN decoder.
In summary, we make the following key contributions: Encoding Mechanism. We propose novel encoding approach to integrate partial observations with spatial coordinate information. effectively encoding these features into a customized geometric grid. @ Novel Methodology. We introduce multi-scale Graph ODE module to model continuous-tin -imm dynamics, complemented by neural auto-regressive correction module to assist and constrain the parametric evolution, ensuring robust and accurate predictions. & Superior Performance. Our method demonstrates state-of-the-an performance on complex synthetic and real-world datasets, showcasing the possibility for accurate and efficient modeling of intricate dynamical processes and precise long-term predictions.
Related Work
2.1
Deep Learning for Physical Simulations
Recent advancements in deep neural networks have established them as effective tools for addressing
the complexities of dynamics modeling [15, 52, 50], demonstrating their ability to efficiently capture the intricacies of high-dimensiooal systems. Physics- Inffrres-n Neural Networks (PINNs) [21, 44, 25]. which optimize weights by minimizing the residual loss derived from the PDE, have received considerable attention due their flexibility in tackling wide range of data-driven solutions [27, 46). Recently, neural operators [24, 29, 49, 35] map between fiiniteedimensional function spaces by replacing standard convolution with continuous alternatives. Specifically, they utilize kernels in fourier space [29, 28, 42] or graph structure [30, 54, 57, 14] to learn the correspondence from the initial condition to the solution at a fixed horizon. However, most existing methods are limited by static observation grids, restricting their ability to evaluate points outside training grids and

---

Tupatts
Curlomsteed
Multfissale Graph ODE Modile
Qirry
Multi-sale Gnaph
Continauu
Diseree Tines
D(R(F()))
Dwooder
Eontreddings
Predintt
Groved muth
x1.
Inttial stote
Louttion Envbulding
A Eellttts
Neural Auto-corvcton Modale
Figure 2: The overview of CoPS. Stage I: Employ multiplicative filter network to encode the initial representation, and map it to the customized grids with message passing scheme. Stage 2: Model the latent dynamics with multi-scale Graph ODE module and auto-correction module in a continuous- time way. Stage 3: Extrapolate results for arbitrary future time step and coordinates.
confining queries to fixed time intervals. this work, we aim to overcome discretization and achieve spatiotemporal modeling in continuous way.
2.2 Advanced Space-Continuouu Modeling
Interpolation has been widely adopted in numerous applications [6, 20, 47], which also holds wide prospects in spatiotemporal modeling. For instance, researchers employed interpolation as post- processing technique generate continuous predictions. Recently, MAgNet [5] proposes predict the dynamical solution after interpolating the observation graph in latent space to new query points. Another approach for space-continuout neural networks, called Implicit Neural Representations (INRs) [17, 23J. Typically. INRs take the spatial coordinates as inputs along with other conditions, which can utilized to enhance real space-continuous modeling and dynamical evolution predicting. DINO [52] overcomes the limitation of failing to generalize to new grids or resolution via a continuous dynamics model of the underlying flow. MMGN [32] learns relevant basis functions from sparse observations obtain continuous representations after factorizing spatiotemporal variability into spatial and temporal components.
priddindependence modeling is kind of coordinate-base
Methodology
Problem Definition. In this work, we focus on modeling spatiotemporal dynamical systems through partial observations. The systems are defined over the temporal domain and spatial domain Ω. The observations recorded at arbitrary consecutive time points f1.L : (tI....,ty t.) E T. and N arbitrary local sensor measurements general real-world scenarios, assume that all dynamics are Iearned from the initial observation u(ta) = (u(x1, fu)................... u(xN: to))². Moreover, the values of time steps and observation locations may vary across different observed trajectories during the inference phase. Thus, for prediction, goal is to learn a neural function Q(-). which maps the initial observation at the first time step to future dynamical predictions arbitrary time steps t; and locations Izj.
locations x, E Q = ( To adapt to
Q(u(to, zo): T.) + Vtt,, t, E 2, E Q.
(1)
Framework Overview. In this section, we introduce the proposed CoPS, which achieves the continuous spatiotemporal modeling from partial observations within complicated dynamical systems. We aim to simultaneously overcome the discretization limitations in both the spatial and temporal domains. To better align with the form of numerical solutions, begin with partial observations of the initial state and apply constraints based on real observations during the continuous evolution process. An overview of our CoPS be seen in Figure 2.
3.1 Initial Encoding via Multiplicative Filter Network
Initial Encoder. Further, we follow the multiplicative filter network and choose nonlinear Gabor filter [10] (-) to generate continuous global frequency representation. The formulation is as

---

follows:
((() 2
|
11)) sin (w:()) x + b(t)).
(/))
9N (z) exp
where u(()) E dR and 7(()) E IR xd, denote the respective mean and scale term of the s-th Gabor filter. Subsequently, the filters are multiplied by the linear transformation of previous layer's embedding and the historical sequence representation ug The iteration process is defined as follows:
P 91 (1a)
7 Pi
(x+1) (u: + W A + b() + gh (I), for K. = ,,...,M (u, 2.) u, Wi P3 + 6,M,
where denotes element wise multiplication, W and by are the learnable parameters. Finally, (u,, is the obtained vector that contain coordinate information and feature of node i.
Customized Grids and Message Passing Scheme. For our observational data, the spatial domain is non-discrete. Recent works like MeshCraphNNet [34] rely on fixed meshes, which present challenges when generalizing to previously unseen coordinate points. We propose novel customized grid mapping approach. Specifically, for any arbitrary coordinate, it can be mapped onto uniform grid structure. Let P; (I,, 2/) denote arbitrary point in the continuous 2D space, we will discover the appropriate grid cell according to it. enrich representation, we connect each point p; to four vertices {121, Vil, U3, 144) of the associated grid cell.
Based on the encoded initial state # = Hu (u;, (u;,I:) æ..) and the customized grids, the next step involves message-passingg scheme that propagates the features from the original continuous space to the customized grids. To further preserve spatial continuity, the scheme encodes and incorporates the relative positions of the connected points during message passing. The transformation is as follows:
h h
Σ jEN(p.)
(ng -h + +0()) b",
where W and 5 are learnable parameters, and d( x;) denotes the relative position embedding of node and j. In this way, we collaboratively encode both the feature information and rich spatial information onto the customized grids, completing the initial encoding process.
3.2 Latent Dynamics Modeling
Further, we view high-dimensional features the customized grids latent states and aim to efficiently model complicated dynamics in a continuous-time way.
Multi-scale Graph ODE. To capture spatiotemporal dynamics over complex domains, we build hicrarchical graph representation inspired by GraphCast [26] on top of the customized grids.
Our objective is to progressively encode both coarse global behaviors and refined local structures. Concretely, let {] be a collection of graphs, where ( coincides with the finest-scale grid obtained from the initial encoding process. Specifically, within each graph g(+), we connect each node to its spatial neighbors by "jump adjacency" on the underlying 2D grid, thus g(2)....,g? g(S) providing graph topologies with longer-range dependencies.
To perform hierarchical message passing, we process each scale independently and then fuse the features using an attention mechanism. For each scale s, the message passing is defined as follows:
h (1.s)
AGGREGATE ((N.) (N :j (EN
n (1,>) COMBINE
(n. (n. (1-1.)) nn (1.6)) (5)
where (i) denotes the neighbors of node at s-th scale. After propagating messages within each scale, employ an attention mechanism to fuse the features across scales.
(1) = Σ
(s) (laa) a (1.a))
ag (W (Wqurryh(l.e)) * (Wreyh? (--)).
(6)
where (*)))) denotes the attention weight, W Wquery and Wkey are learnable parameters, and * denotes the cosine similarity computation. Building on this hierarchical representation, we parametrize coupled Graph ODE to model the continuous- time evolution of it's latent state.
dh: & (hi, h/. .. w.w. G) dt
(7)

---

capture complex corrections at selected Here, & denotes the multi-scale message passing function, G represents the hierarchical graph structure, and encapsulates the model parameters. Given the ODE function 4, the node initial values, and the corresponding contextual vector, the latent dynamics can be solved by any ODE solver like Runge-Kuut Kutta [37].
..... hT ODESolve(D, [4] h2 11
(8)
This formulation allows the model to theoretically obtain the hidden state at any arbitrary timestep, provided that the Neural ODE fits the underlying dynamics well.
Neural Auto-correction. Although the multi-scale Graph ODE framework provides a global continuous- time evolution, it still relies the strong assumption that learnable ODE exists to model the concrete physical dynamics. Moreover, neural ODEs relying on numerical solvers may fail to capture the system's 's intrinsic nonlinear features, making difficult to handle error divergence during evolution. To address this, we introduce a neural auto-correction module that imposes a discrete dynamical regime in the latent space. Thus time steps in a heuristic Markov chain manner.
At each correction step, convolutional encoder first compresses the current latent representation into a compact state. Subsequently, after the discrete evolution block, transposed convolution-basee decoder reconstructs the refined latent field. This operation acts "Markov state observer" [8], learning a discrete single-step mapping in a more compact latent space. The formulation is as follows:
E(-) = o (L)(Con2d(()))) D(-) = Tanh ( (UnConv2d(-))
(9)
Further, between the encoder and decoder lies neural block that embodies a discrete transition operator similar to a Markov kernel. Technically, it consists a 1 Conv2D, which strips away extraneous channels to focus on the most critical latent features. Then, it is followed by parallel 2roupComv2D [41] operators to process these features in multiple subspaces. We utilize R to represent the combination of the x Conv2D and the parallel GroupComv2D. Then the whole evolution can be formulated as:
1k+ = D(R(E(
(10)
where =(k) denotes the latent embedding at discrete step k. Crucially, this transformation depends only on the current embedding =(k), thus fulfilling a Markov property at each correction step. By encapsulating key spatiotemporal features in z(k) and evolving them into 2(k: + 1) via R(-), the neural auto-correction module adaptively regularizes the global continuous-time solution. On one hand, it provides a local perspective on how errors can be contained and corrected in each interval. On the other hand, it retains sufficient memory of the system's 's evolving states without requiring full historical trajectories.
Inference Strategy. During inference, we employ an iterative strategy that combines the multi- scale Graph ODE module and the neural auto-correction module to generate continuous accurate predictions. Starting from the initial state h42(o),, the model proceeds as follows:
Continuous Evolution: The multi-scale Graph ODE module evolves the latent states continuously from ty to fx+1:
(/me (la(1), (l(()))N)))))) di.
he(1e41)= he(lx)
(11)
Discrete Correction: At each discrete time step 1++1, the neural auto-correction module refines the predicted state:
1 (f = (+(+++1) + Xr (h+(+))), where rg denotes the neural auto-correction module, and À is the hyperparameter for balance. The corrected state h (1x+1) is used as the initial state for the next time step, and the process repeats until the final prediction time is reached. This inference strategy ensures that the model can generate accurate and stable predictions over long time horizons, while maintaining the continuity and smoothness of the learned dynamics.
(12)
3.3 Decoder
Upon obtaining the latent states on the customized grids, we project them back the original continuous spatial domain for final predictions. Specifically, for a query coordinate gm {z Fo
5

---

we first identify the corresponding grid cell Um and establish connections to its four vertices, denoted {Uml, Um2 9m3. Wm4). We then initialize the representation of 9m with single step Gabor filter transformation Imm. 9i (fm)) ), and then perform message-passinng -passing update just as in Eq (4). Further, we can obtain the corresponding predictions through 2-layer MI.P decoder D(.).
3.4 Theoretical Analysis
Theorem 3.1 (Error Bounding via Hybrid Continuous-Diserve Latent Corrections). Consider spatiotemporal dynamical system whose latent representation y(t) intended to follow an ideal trajectory y (t). The learned dynamics are modeled by a Neural ODE:
d dt v()) F(y(t).
(13)
with initial condition y(ta). The function F, representing the multi-scale graph message passing, is assumed to be Læ-Lipschitz continuous with respect to y(t). Periodic corrections are applied at discrete time steps l = to + k. Abcore- Let y(tx) ) be the state before correction and y(t²) ) be the state after applying a neural auto-correction operator Cup:
v((t) v((T) + Co(v(()))
(14)
The error is defined as e(t) = Ily (t) - (())l. We assume:
(a) The combined error from the ODE model discrepancy (vis-à-vis (t)) and numerical solver inaccuracies over one interval Aicor is bounded by EODE, such that e(tx+1) S Ly Staare e(ttt) + EonE.
(b) The correction operator Cyp reduces the error proportionally and introduces a bounded residual, i.e., e(lit) S K. e((T)) + 8c, where S k < a contraction coefficient and dc 2 O is base error from the corrector.
Then, for K L(1 - 10)/ Aloor] correction steps, if the effective error amplification per cycle Ceff kg Meore < 1, the error at time I. z ta + K Mccorr is bounded by:
|e(())|| s off (c(tu)) 1 1 1-ael aclf
KEODE + de ) KEODE + 0c
(15)
This can be expressed more generally as:
|ee(t)|| s C1 aell
((t-to)/ + C22
(16)
where C₁ = =m,x(0,, e(to)-Co) or simply e(to) ((fo))oo) for looser bound, C2 = KE is the asymptotic I-aut
error floor, and E 10, quantifies the error decay rate per correction cycle, dependent on the system's Lipschitz constant, the correction interval, and the corrector's efficacy. This bound demonstrates that the error converges to a non-zero floor C2 if Creft < 1.
4 Experiment
4.1
Experimental Settings
To comprehensively illustrate the property and efficacy of the extrapolations obtained from CoPS, we conduct experiments on diverse synthetic and real-world datasets. + For synthetic datasets, we first choose Navier- [29] and by numeric PDE solvers. Then, we select Prometheus [48], which is a large-scale combustion dataset simulated with industrial software. + For real-world datasets, we choose WeatherBench [36], dataset for weather forecasting and climate modeling. We also select Kuroshio [49], which provides vector data of sea surface stream velocity from the Copernicus Marine Service. See Appendix A for detailed deseriptions of all these datasets.
Datasets.
Bénard Convection [43], which are directly generated
Baselines. We evaluate our model against three baselines representing the state-of-the-art in continu- ous modeling. MAgNet [5] employs an "Encode- Sateeeeetteetteeeooooooooe MAgNet employs the nearest neighbor interpolation technique to generalize to new query points.
scheme. Specifically,

---

Subsequently. it forecasts the evolutionary trends by leveraging GNN-based message passing neural network. DINO [52] learns PDE's flow to forecast its dynamical evolution by leveraging spatial implicit neural representation modulated by a context vector and modeling continuous-time evolution with a learned latent ODE. This is the closest approach to our method. ContiPDE [39] formulates the task as a double observation problem. It utilizes recurrent GNNs to roll out predictions of anchor states from the IC, and employs spatiotemporal attention observer to estimate the state at the query position from these anchor states. We utilize the official implementation for all models and tune their experimental settings to follow the requirements of our tasks.
Tasks. We assess the performance of CoPS across diverse forecasting tasks to evaluate their efficacy in various scenarios. We use the mean squared error (MSE) as the performance measurement. (i) Sparse Flexibility Gauging the effectiveness of predicting global field evolution based on observations with diverse degrees of sparsity (subsampling ratio of 25%, 50%, 75%). (ii) Time Flexibility Evaluating our model's 's performance in extrapolating beyond the training horizon. Here we design two experimental setups: long-term extrapolation beyond the training horizon, and continuous-time prediction for intermediate points between discrete time steps. (iii) Resolution Generalization Investigating the performance of generalizing to new resolution (up or down). (iv) Super-resolution Investigating the model's effectiveness of super-resolution query and extrapolation. (v) Noise Robustness Exploring the robustness of our model against varying noise ratios. (vi) Ablation Study Investigating contributes of each key component to the performance of CoPS.
Implementation. For synthetic data like Stoker-Stokes, the train and test sets differ only by their initial conditions. The samples are partitioned in a 7:2:1 ratio into training. validation, and test sets. For real-world data like WeatherBench, we use the historical ERAS global atmospheric reanalysis data. The data was partitioned strictly by date to prevent any data leakage from the future into the training process. Specifically, the train set uses data from 1979-2018, the validation set from 2019, and the test set from 2020-2022. To ensure faimesss we use partial observations of the initial state as input and supervise the training with future real observations at discrete time steps (only using observed future values). In evaluating spatial continuity, we perform subsampling of the full initial state at rates of (25%, 50%, and 75%). ). assessing both observed and unobserved points. For temporal continuity, we evaluate three criteria based on different subsampling rates: predictions within the training horizon, long-term predictions beyond the training horizon, and continuous time predictions for intermediate points between discrete time steps. More details are illustrated in Appendix C.
4.2
Main Results
Space Flexibility. To evaluate the spatial querying ability of our method, we adjust the number of available measurement points by using different proportions of observation points (25%, 50%, 75%) in the training data. We report the MSE compared to the ground truth on four datasets, as shown in Table From the table, we see that our method (Ours) significantly outperforms the baseline methods on all datasets and at all observation sparsity levels. On the Navier-Stoke dataset, with only 25% of the observation data, our method achieves MSE of 2. 2.944E-03 and 5.743E-03 under the In-s and Ext-s settings, which are much better than ContiPDE (5. 15.456E-03 and 9,523E-03) and DINo (1.074E-02 and 2.537E-02). As the observation ratio increases to 50% and 75% our model's performance further improves. The MSE for In-s decrease to 2.253E-03 and 1 1.464E-03, and for Ext-s decrease to 4.571E-03 and 2.872E-03. On the Kuroshio dataset, even with only 25% observation data, our method achieves MSE of 1.285E-03 and 2.381 under In-s and Ext-s, significantly better than other methods. For example, ContiPDE's MSE are 2.352F-03 and 3. 3.763E-03, and DINo's 's are 3. 3.737F-03 and 5.923F-03. As the observation ratio increases, our method's 's MSE further decreases, and its accuracy becomes higher. On the Prometheus and WeatherBench datasets, our method also performs excellently. Especially on the WeatherBench dataset, when the observation ratio is 75%, our method achieves MSE of 3.585E-03 and 8.472E-03 under In-s and Ext-s, much lower than other baseline methods. These results show that our method effectively recovers the global scene from partial observations on different datasets and observation sparsity levels. It demonstrates robustness in handling sparse data and spatial generalization.
Time Flexibility, To evaluate our method's performance in temporal extrapolation, we train and evaluate the model with observation subsampling ratio of 50% and 100%. We use three extrapolation modes: (1) Extrapolation within the discrete training range (In-t); (2) Extrapolation beyond the discrete training range (Ext-t); (3) Extrapolation at intermediate points between time steps (Con-t).
7

---

Table 1: To evaluate the spatial inquiry power of our method, we vary the number of available measurement points in the data for training from 25%, 50% and 75% amount of observations. We report MSE compared to the ground truth solution.
Moner
NAVIE--TOOIS
KEROSHIO
PROVITHRUS
WEATHORRENCH
3.1645-02
2.7758-02
2.3055-02
7:104E:03
6:5338-03
1,448-02
L.233-022
3.635E
3.028E-02
2.4838-02
Exns
5.846B-02
4.2856-00
3.1146-00
9.045:00
8.1138 03
7.5186-03
2.7756-00
2.3236-02
1.7245-02
33600000
4.9726-02
3.2958-02
DINO
Bxns
16.5
2337B-42
1,074--02
9.564E-08
1,7618-00
7.5546.00 03
9.2485-03
3.7378.03
19356-03
5 1.3110-03
3.3328.0 03
2.8545 03
43776-03
1.2236-02
1,7815-00
8 3.0056-03
1,3465-02
5.65TE 03
8,3745-00 03
38338-02
2.365 02
2. 2.766-03
2.186E-02
1.9226022
7 2.5880-42
CONDPDE
18-5
5.4585455
4.1768-05
1.8258-40
1:352-001
1.9478-03
1.503 01
9.4626-05
6.0948-03
4.1632.00 03
1.5428-022
1.294--02
1.058--22
9,5236-45
1.9758-06
6.2318/08
3.753E:03
2:5318:03
1J225-022
5.3558.00
2.1726-02
1.63E-02
14746-02
29944-43
1.4648-03
1.2959-03
5.1768:63
5.249E-43
3.565E-43
5.7436-83
4.5718-03
2.8728-03
1,381E-03
1.577E-43
9.2728-44
9.2648:03
5.3428:00
4.3878:03
1.769E-02
1.050--03
9.4726-43
G Imputs
--2%%
1-15%
Inpure
Our method in detailed
DoNe
Dnn
GroundTrruth
Grosnd-Truth
Predictio
Predictio
Pradicioo
p-10
Gmund-Truth
Inpuus
x725%
A-00%
(d)
Impure
Oree
ComIPDg
Dina
(6)
0m03
DINo
0.000
Ground
Predictoo
Prodiction
Prodiction
Guound Truth
Eroor
Einor
Esro
Outs
10
50
Prodictio Step
Local Details
Figure 3: Figure(a) and (b) shows the perdiction performance based on observations with diverse ratio of subsampling (25%, 50%. 75%) on the Navier-Stokes and Kuroshio dataset. Figure(c) demonstrates long-term extrapolation beyond the training horizon on the Prometheus dataset, and Figure(e) shows the evolution of prediction errors over time steps, revealing the increasing error with longer prediction steps. Figure(d) illustrates continuous-time prediction for intermediate points between discrete time steps on the WeatherBench dataset.
We report the mean squared error (MSE) on the Navier-Stokes, Beyleigh--éénadd Convection, and Prometheus datasets, as shown in Table 2. From Table 2, we see that our method achieves the best performance on mostly all datasets and extrapolation settings. For example, the Navier- Stokes dataset with a subsampling ratio of 50%, our method achieves MSE of 2.832E-03 (In-t), 5.764B-03 (Ext-t), and 1355-00 (Con-t). These are significantly better than ContiPDE (5.054E-033, 8.342E-03, 6.4 6.447E-03) and DINo 1E-03, 1.074E-02, 9.971E-03) When the subsampling ratio increases 100%. the performance improves further. Our method's MSE decrease 1.132E-03 (In-t), 2.364E-03 (Ext-t), and 1.735E-03 (Con-t). On the Mayleigh-Bénare Convection dataset, our method also achieves MSE of 6 6.522E-04 (In-t) and 1.327E-03 (Ext-t), significantly better than other baseline methods. This shows that our method can make accurate predictions within the discrete training range. It can also effectively extrapolate time points beyond the training range. When we perform continuous-ting time prediction at intermediate points between time steps (Con-t), our method also performs excellently. It achieves the smallest prediction error. Overall, these results demonstrate the strong capability of our method in continuous-time modeling. It provides accurate and reliable predictions under different temporal extrapolation settings.
Visualization and Analysis. Figure provides compelling visual evidence for the superior per- formance of our proposed CoPS across range of challenging tasks. Specifically. Figure (a) and (b) demonstrate that CoPS can effectively model the entire physical field and Iearn the evolution trends of unobserved points only with partial observations. When the subsampling ratio reaches 75%, our model can achieve precise prediction on Navier-Stokes and Kuroshio datasets. As observed in Figure (c) and (e), our method significantly outperforms DINO and ContiPDE in long-term forecasting on the Prometheus dataset by effectively suppressing error accumulation, thereby main- taining high-fidelity details where baselines exhibit blurring. This robust temporal modeling is further evidenced in Figure (d), where CoPS accurately predicts intermediate states between discrete time steps. This showcases its ability to perform true, dynamically consistent integration rather than simple interpolation, confirming its superior continuous-time modeling capabilities.
8

---

Table 2: We evaluate the temporal extrapolation performance of method. Models trained and evaluated with observation subsampling ratio of 50% and 100%. We employ three kinds of extrapolation, containing: (I) extrapolation within discrete training horizon (In-t); (2) extrapolation exceeding discrete training horizon (ExI-)); and (3) extrapolation of intermediate points between time steps (Con-t). We report MSE compared the ground truth solution.
NAVIER- STOKES
NAVIER-STOKES
RAYLEICH BENARD CONVECTION
PROMETHEUS
IN-T
EXT-T
CON-T
IN-T
EXT-T
CoN-T
IN-T
EXP-
CON-T
50% zubrompping ratio
MAGNET
1.253E-02
2.601E-02
1.335E-02
1.138E-02
L824E-02
DINo
7.651E-03
1.074E-02
9.971B-03
5.748E-03
8.472E-03
7.342E-03
8.321E-03
L.117--02
8.852E.03
CONTIPDE
5.054E-03
8.342E-03
6.44E--03
3.74-E-03
3.814E-03
2.908E-03
5,435E-03
B.154E-03
6.637E-03
OURS
2.832E-03
5.764K-03
4.135E-00
1.526K-03
2.3288-003
1.765E-03
3.374E-03
6.678E-00
5.355K-03
100% suhsampling razio
MAGNET
7.429E-03
1.273E-02
4.837E-03
1.235E-03
7.938E-03
L.02EB002
DINo
4.3526-03
7.3748-03
5,2718503
2.2488-03
4.872E-03
3.742E-03
5,7218-03
8,217H-03
6,2888-03
CONTIDDE
2.653E-03
4.742E-03
3.847E-03
1,24-E-03
3.214E-03
2.308E-03
3.835E-03
6.554E-03
4.037E-03
OURS
1.132E-03
2.364F-03
1.739E-03
6.522R-04
1.3277E-03
2.877F-03
8.174E-03
P P
Greeslinc
120-322
236-256
Premschop
0.300
WeatherBeach
MAght
Might
DEN:
DPh
0.333
Oata
Preditie
Prodedus
Prokites
00,00
Stvie--tokes
0201311
0.322
Rovalation
Revalatten
Figure 4: Left shows the inference performance for super-resolution on Navier-Stokes dataset. Right demonstrates the resolution generalization capability compared with MAgNet, DINo, and ContiPDE.
4.3 Model Analysis
Analysis Super-resolution. In the super-resolution experiment presented in Figure 4 (left), the results illustrate the inference performance on the Navier-Stokes dataset across resolutions of 32 x 32, 64 x 64, 128 128, and 256 x 256. The results consistently demonstrate improved prediction quality resolution increases. Compared to baseline models (MAgNet, DINo, and ContiPDE), the proposed model effectively reduces prediction error at high resolutions, maintaining a lower MSE and providing clearer, more accurate predictions of the original Navier-Stokes patterns.
Resolution Generalization. As observed in Figure 4 (right), the folding diagrams depict the resolution generalization capabilities for both Prometheus and WcatherBench datasets. The proposed CoPS consistently outperforms the comparative models (MAgNet, DINo. ContiPDE) at all resolution levels, achieving the lowest MSE value. This result highlights CoPS's robustness and effectiveness in handling resolution variations. The findings emphasize the model's ability to generalize across different resolutions, which is crucial for applications requiring detailed and precise predictions in complex dynamic systems.
Ablation Study. To evaluate the contri- bution and importance of each component in the proposed CoPS, we conduct corre- sponding ablation experiments, and use MSE error as metric. Our model variants follows: CoPS w/o MFN, we re- move the multiplicative filter network and use an MLP. CoPS w/o MGO, we remove the multi-scale Graph ODE module. O CoPS w/o NAC, we remove the neural auto-correction module. Table 3 shows the comprehensive results of our ablation experiment. The results of the ablation experiments show that removing any component results in decrease in predictive performance, further proving the critical role of these components in CoPS framework. Specifically, the performance declines after removing the multi-scale Graph ODE module is particularly evident, proving its importance in capturing internal dynamic representations.
Table 3: Ablation study on Kuroshio, (Report MSE)
IN-S /I--T
Ext-s IN-T
IN-S Ext-T
Ext-s EXI-T
w/o MFN
1.027E-03
1.946I-03
2.458E-03
3.149E:03
w/o MGO
1.594E-03
2.357E-03
3.053E-03
4.175E-03
1.428F-03
2.216F.03
2.742F-03
3.657F-03
OURS
7.253E-04
1.577K--3
2.161E-03
2.9388-00

---

Conclusion
In this paper, we introduce CoPS, a novel data-driven framework for modeling continuous spatiotem- poral dynamics from partial observations. To overcome discretization, we first customize geometric grids and employ multiplicative filter network to fuse and encode spatial information with the corre- sponding observations. After encoding to the concrete grids, CoPS model continuous-tima dynamics by designing multi-scale graph ODEs, while introducing a Marcov-based neural auto-correction module to assist and constrain the continuous extrapolations. Extensive experiments both synthetic and real-world datasets demonstrate the superior performance of CoPS, which underlines the potential to provide robust method for accurate long-term predictions in face of sparse and unstructured data.
Acknowledgement
This work was partially supported by the NSFC Program (No. 62202042, No. 6212780016); in part by
the Supported by the Fundamental Research Funds for the Central Universities (No. 2024JBMC031); in part by the Aeronautical Science Foundation of China (No. ASFC-2024Z071 10M5002; in part by the by the OpenFund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province(No.
SELACSS-2023111 the Guangdong S&T Programme (No. 202480101030002) the Ministry of Industry and Information Technology of China, the National Key Research and Development Program of China (No. 2023YFB3307500 and the Science and Technology Innovation Project of Hunan Province (No. 2023RC4014).
References
[1] Md Ashiqur Rahman, Zachary E Ross, and Kamyar Azizzadenesheli, U-no: operators. arXiv e-prints, pages arXiv-2204, 2022.
U-shaped neural
[2] Timothy Barth, Raphaèle Herbin, and Mario Ohlberger. Finite volume methods: foundation and analysis. Encyclopedia of computational mechanics second edition, pages 1-60, 2018.
[3] Marin Biloš, Johanna Sommer, Syama Sundar Rangapuram, Tim Januschowski, and Stephan Günnemann. Neural flows: Efficient alternative to neural odes. Advances in neural information processing systems, 144-13255-2133 2021.
[4] Boris Bonev, Thorsten Kurth, Christian Hundt, Jaideep Pathak, Maximilian Baust, Karthik Kashinath, and Anima Anandkumar. Spherical fourier neural operators: Learning stable dynamics the sphere. In International conference machine learning, pages 2806-28233 PMLR, 2023.
[5] Oussama Boussif, Yoshua Bengio, Loubna Benabbou, and Dan Assouline. Magnet: Mesh agnostic neural pde solver. Advances Neural Information Processing Systems, 35:31972- 31985, 2022.
[6] Cristian Challu, Kin G Olivares, Boris Oreshkin, Federico Garza Ramirez, Max Mergenthaler Canseco, and Artur Dubrawski. Nhits: Neural hierarchical interpolation for time series fore- casting. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pages 6989 6997, 2023.
[7] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
[8] Krzysztof Choromanski, Valerii Likhosherstov, David Dohan, Xingyou Song, Andreea Gane, Tamas Sarlos, Peter Hawkins, Jared Davis, Afroz Mohiuddin, Lukasz Kaiser, et al. Rethinking attention with performers. arXiv preprint 140994 2020.
[9] Michaël Defferrard, Martino Milani, Frédérick Gusset, and Nathanaël Perraudin. Deepsphere: graph-based spherical cnn. In International Conference Learning Representations, 2020.
[10] Rizal Fathony, Anit Kumar Sahu, Devin Willmott, and Zico Kolter. Multiplicative filter networks. In International Conference Learning Representations, 2020.
10

---

[11] Zaige Fei, Fan Xu, Junyuan Mao, Yuxuan Liang, Qingsong Wen, Kun Wang, Hao Wu, and Yang Wang. Open-CK: A large multi-physics fields coupling benchmarks in combustion kinetics. In The Thirleenth International Conference on Learning Representations, 2025.
[12] Katerina Fragkiadaki, Pulkit Agrawal, Sergey Levine, and Jitendra Malik. Learning visual predictive models of physics for playing billiards. arXiv preprint orXiv:/51100440) 2015.
[13] RH Gallagher, JT Oden, C Taylor, and OC Zienkiewicz. Finite element. Analysis: Fundamentals. Prentice Hall, Englewood Cliffs, 1975.
[14] Yuan Gao, Hao Wu, Ruiqi Shu, Huanshuo Dong, Fan Xu, Rui Ray Chen, Yibo Yan, Qingsong Wen, Xuming Hu, Kun Wang, et al. Oneforecast: universal framework for global and regional weather forecasting. arXiv preprint arXiv:2502.00333, 2025.
[15] Zhihan Gao, Xingjian Shi, Hao Wang, Yi Zhu, Yuyang Bernie Wang, Mu Li, and Dit- Yan Yeung. Earthformer: Exploring space-time transformers for carth system forecasting. Advances in Neural Information Processing Systems, 35:25390-25400 2022.
[16] Nicholas Geneva and Nicholas Zabaras. Modeling the dynamics of pde systems with physics- constrained deep auto-regressive networks. Journal of Computational Physics, 403:109056, 2020.
[17] Daniele Grattarola and Pierre Vandergheynst. Generalised implicit neural representations. Advances in Neural Information Processing Systems, 35:30446 30458, 2022.
[18] Zhenyun Hao, Jianing Hao, Zhaohui Peng, Senzhang Wang. Philip S Yu, Xue Wang, and Jian Wang. Dy-hien: Dynamic evolution based deep hierarchical intention network for membership prediction. In WSDM, pages 363-371, 2022.
[19] Valerii lakovlev, Markus Heinonen, and Harri Lähdesmäki. Learning continuous- pdes from sparse data with graph neural networks. arXiv preprint arXiv:2006.0892 2020.
[20] Valerii lakovlev, Markus Heinonen, and Harri Lähdesmäki. Learning space-time continuous latent neural pdes from partially observed states. Advances in Neural Information Processing Systems, 36, 2024.
[21] Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. Variational physics-informed neural networks for solving partial differential equations. arXiv preprint arXiv:1912.008: 2019.
[22] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv: (oXiv:66090029000 2016
[23] David Knigge, David Wessels, Riccardo Valperga, Samuele Papa, Jan-Jakob Sonke, Erik Bekkers, and Efstratios Gavves. Space-time continuous pde forecasting using equivariant neural fields. Advances in Neural Information Processing Systems, 37:76553-76577 2024.
[24] Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research, 24(89):1-99 -97, 2023.
[25] Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in thysicc-iinfrmeed neural networks. Advances in Neural Information Processing Systems, (166588-26666) 2021.
[26] Remi Lam, Alvaro Sanchez- Gonzalez, Matthew Willson, Peter Wirnsberger Meire Fortunato, Ferran Alet, Suman Ravuri, Timo Ewalds, Zach Faton-Rosen, Weihua Hu, et al. Learning skillful medium-range global weather forecasting. Science, 382(6677):1416--4222 2023.
[27] Jun Li, Gan Sun, Guoshuai Zhao, and H Lehman Li-wei. Robust low-rank discovery of data- driven partial differential equations. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 767-774, 2020.

---

[28] Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural oper- ator with learned deformations for pdes on general geometries. arXiv preprint arXiv:2207.05209 2022.
[29] Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differen- tial equations. arXiv preprint arXiv:2010.08895, 2020.
[30] Zongyi Li, Nikola Kovachki, Kamyar Azizzadeneshhlli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Graph kernel network for partial differential equations. arXiv preprint ar XXiv:2003.03485, 2020.
[31] Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang, Hiroaki Hayashi, and Graham Neubig. Pre-train, prompt, and predict: systematic survey of prompting methods in natural language processing. ACM Computing Surveys, 55(9):1-35, 2023.
[32] Xihaier Luo, Wei Xu, Yihui Ren, Shinjae Yoo, and Balu Nadiga. Continuous field reconstruction from sparse observations with implicit neural networks. arXiv preprini arXiv:2401. 11611, 2024.
[33] Mohammed S Mechee and Sameeah H Aidi. Generalized euler and runge-kutta methods for solving classes of fractional ordinary differential equations. International Journal of Nonlinear Analysis and Applications, 13(1): 1737-17777 2022.
[34] Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter W Battaglia. Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409, 2020.
[35] Bogdan Raonic, Roberto Molinaro, Tim De Ryck, Tobias Rohner, Francesca Bartolucci, Rima Alaifari, Siddhartha Mishra, and Emmanuel de Bézenac. Convolutional neural operators for robust and accurate learning of pdes. Advances in Neural Information Processing Systems, 36, 2024.
[36] Stephan Rasp, Stephan Hoyer, Alexander Merose, Ian Langmore, Peter Battaglia, Tyler Russel, Alvaro Ganzaez- Vivian Yang, Rob Carver, Shreya Agrawal, et al. Weatherbench 2: A benchmark for the next generation of data-driven global weather models. arXiv preprint arXiv:2308. 15560, 2023.
[37] Michael Schober, David K Duvenaud, and Philipp Hennig. Probabilistic ode solvers with runge-kutta means. Advances in neural information processing systems, 27, 2014.
[38] Suihong Song, Tapan Mukerji, and Jiagen Hou. Bridging the gap between geophysics and geology with generative adversarial networks. IEEE Transactions on Geoscience and Remote Sensing, 60:1-11, 2021.
[39] Janny Steeven, Nadri Madiha, Digne Julie, and Wolf Christian. Space and time continuous physies simulation from partial observations. In ICLR, 2024.
[40] Rob Stoll, Jeremy A Gibbs, Scott T Salesky, William Anderson, and Mare Calaf. Large-eddy simulation of the atmospheric boundary layer. Boundary-Layer Meteorology, 177:541-581, 2020.
[41] Cheng Tan, Zhangyang Gao, Lirong Wu, Yongjic Xu, Jun Xia, Siyuan Li, and Stan Z Li. Temporal attention unit: Towards efficient spatiotemporal predictive learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18770-18782, 2023.
[42] Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. In ICLR, 2023.
[43] Rui Wang, Karthik Kashinath, Mustafa Mustafa, Adrian Albert, and Rose Yu. Towards physics- informed deep learning for turbulent flow prediction. KDD, 2019.
[44] Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neural tangent kernel perspective. J. Comput. Phys., 2020.
12

---

[45] Weiyan Wang, Xingjian Shi, Ruiqi Shu, Yuan Gao, Rui Ray Chen, Kun Wang, Fan Xu, Jinbao Xue, Shuaipeng Li, Yangyu Tao, al. Beamvq: Beam search with vector quantization to mitigate data scarcity in physical spatiotemporal forecasting. arXiv preprint arXiv:2502. 18925, 2025.
[46] Hao Wu, Yuan Gao, Ruiqi Shu, Zean Han, Fan Xu, Zhihong Zhu, Qingsong Wen, Xian Wu, Kun Wang, and Xiaomeng Huang. Turb-11: Achieving long-term turbulence tracing by tackling spectral bias. arXiv preprint 2XXi:25505.1933 2025.
[47] Hao Wu, Changhu Wang, Fan Xu, Jinbao Xue, Chong Chen, Xian-Sheng Hua, and Xiao Luo. Pure: Prompt evolution with graph ode for outto--distribution fluid dynamics modeling. Advances in Neural Information Processing Systems, 37: 17:109955-009994 2024.
[48] Hao Wu, Huiyuan Wang. Kun Wang, Weiyan Wang. Changan Ye, Yangyu Tao, Chong Chen, Xian- -henn Hua, and Xiao Luo. Prometheus: Out-of dfi-rriititittoinnn fluid dynamics modeling with disentangled graph ode. In Proceedings of the s1 International Conference on Machine Learning, page PMLR 235, Vienna, Austria, 2024. PMLR.
[49] Hao Wu, Kangyu Weng, Shuyi Zhou, Xiaomeng Huang, and Wei Xiong. Neural manifold operators for learning the evolution of physical dynamics. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 3356-3366, 2024.
[50] Hao Wu, Fan Xu, Chong Chen, Xian-Sheng Hua, Xiao Luo, and Haixin Wang. Pastnet: Introducing physical inductive biases for spatio-temporal video prediction. In Proceedings of the 32nd ACM International Conference on Multimedia, pages 2917-2926, 2024.
[51] Fan Xu, Nan Wang, Hao Wu, Xuezhi Wen, Xibin Zhao, and Hai Wan. Revisiting graph-based fraud detection in sight of heterophily and spectrum. Proceedings of the AAAI conference on artificial intelligence, volume 38, pages 9214 -9222, 2024.
[52] Yuan Yin, Matthieu Kirchmeyer, lean-Yves Franceschi, Alain Rakotomamonjy, and Patrick Gallinari. Continuous pde dynamics forecasting with implicit neural representations. arXiv preprint wIXi::2209.14855, 2022.
[53] Yanfu Zhang, Shangqian Gao, Jian Pei, and Heng Huang. Improving social network embedding via new second-order continuous graph neural networks. In KDD, pages 2515-2523, 2022.
[54] Zeyang Zhang, Xin Wang, Ziwei Zhang, Haoyang Li, Zhou Qin, and Wenwu Zhu. Dynamic graph neural networks under spatio-temporal distribution shift. Advances in neural information processing systems, 35:6074-6088 6089, 2022.
[55] Xi Zhao, Linghsuan Meng, Xu Tong, Xiaotong Xu, Wenxin Wang, Zhongrong Miao, Dapeng Mo. et al. A novel computational fluid dynamic method and validation for assessing distal cerebrovascular microcireulatory resistance. Computer Methods and Programs in Biomedicine, 230: 107338, 2023.
[56] David Zheng, Vinson Luo, Jiajun Wu, and Joshua B Tenenbaum. Unsupervised learning of latent physical properties using perception-predicee networks. arXiv preprint 2018.
09244,
[57] Yanping Zheng, Hanzhi Wang, Zhewei Wei, Jiajun Liu, and Sibo Wang. Instant graph neural networks for dynamic graphs. In KDD, pages 2605-2615, 2022.
13

---

NeurIPS Paper Checklist
The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
You should answer [Yes], [No], or [NA].
[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.
Please provide short (1-2 sentence) justificcton right after your answer (even for NA).
The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation.
While "[Yes] " is generally preferable to "[No]", ", it is perfectly acceptable to answer "[No] provided proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] or "[NA] is not grounds for rejection. While the questions are phrased in binary way, we acknowledge that the true answer often more nuanced, so please just use your best judgment and write justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
IMPORTANT, please:
Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",
Keep the checklist subsection headings, questions/answers and guidelines below.
Do not modify the questions and only use the provided macros for your answers.
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification:
The abstract and introduction clearly state the contributions regarding the CoPS framework.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]
14

---

We thoroughly discuss the limitations of our work and propose potential directions for futurr research in Appendix.
Justification:
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "Limitations" section in their paper.
The paper should point any strong assumptions and how robust the results are violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g. if the approach was only tested few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't It acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
Justification:
The proofs of corresponding theorems are provided in Appendix.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification:
We provide detailed workflow and experimental model settings in Appendix.
Guidelines:
15

---

The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors welcome deseribe the particular way they provide for reproducibilityy In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification:
In the abstract, we provide URL link to both the source code.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips. cc/
Beeiii/gguide//oodeueee www.bmmmiiiooooiiccc..... for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https: //n ctt.pss.cc/pubiccuuuuuuu gesds//////eeeeeee/e /CodeSubmian ss ierionnoolliei onPolicy) for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
16

---

Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Question:
Answer: [Yes]
Justification:
We provide experimental settings in Appendix.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification:
The experiments in this paper report variance measurements.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results accompanied by enror bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
OK to report 1-sigma error bars, but one should state The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that out of range (e.g. negative error rates).
If error bars reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We provide information on the computer resources in Experiment section.
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
17

---

The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: //terr//e.....ee wtt.iis...pppppppe cctppblic///ccccccccccccccccclct weabblii//thicsuuiiiii cacuiieleeee
Answer: [Yes]
Justification: All aspects of this work comply with the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.
The authors should make to preserve anonymity (e.g.. if there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification:
We have discussed broader impacts in Appendix.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility ML).
11. Safeguards
Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, scraped datasets)?
Answer: [NA]
Justification: This paper poses no such risks.
Guidelines:
The answer NA means that the paper poses such risks.
18

---

Released models that have high risk for misuse or dual-use use should be released with necessary safeguards to allow for controlled use of the model. for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: We have cited all used public datasets and compared baselines.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g.. website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
This paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moddl part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: This paper does not involve crowdsourcing experiments or research with human subjects.
19

---

Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: This study did not involve human participants, so no risks, disclosures, or IRB approvals were required or obtained.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMS if is an important, original, or non-standard component of the core methods in this research? Note that if the L.M is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: The core ideas proposed in this paper were developed without any involvement of large language models.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components. httprripps.c//ooeeeeee/ee///////
Please refer to our LLM policy for what should or should not be described.
20

---

A Detailed Description of Datasets
Navier-Stokes Equations. Navier-Stokes Equations [29] describe the motion of fluid substances such as liquids and gases. These equations are set of partial differential equations that predict weather, ocean currents, water flow in a pipe, and air flow around wing, among other phenomena. The equations arise from applying Newton's 's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term proportional to the gradient of velocity, and pressure term. The equations are expressed as follows:
au 01
+ u Vu
- -VV++V.TT
0p
+ V (pu) = O,
(17)
01
O(pE) ((pE + p)u) = V (T u) V. (6VT) + pf- u.
Ot
where u denotes the velocity field, P represents the density the fluid, p is the pressure, is the viscous stress tensor, given by (V + (Vu)²) " =u).. E is the total energy per unit mass, E 1 /u/2, is the internal energy per unit mass, T denotes the temperature, and represents the thermal conductivity. All simulations were generated from the Navier-Stokes equation with a constant Reynolds number of le-5, We utilized a total of 1200 independent simulation samples.
Beyleigh-Bénarve Convection. Rayleigh-Bénn Bénard Convection [43] is generated using the Lattice Boltzmann Method to solve the 2-dimensional fluid thermodynamics equations for two-dimensional turbulent flow. The general form of the equations is expressed as:
V v.
Ou + (u. V)u
1
Vp + uv + - T,)]X,
Ot
Po
(18)
OT
Øt + (u V)T = =V
where g is the gravitational acceleration, X is the acceleration due to the body-force of the fluid parcel, Po is the relative density, T represents temperature, To is the average temperature, denotes the coefficient of thermal expansion. and x is the thermal conductivity coefficient. The simulation parameters for the dataset are as follows: Prandtl number 0.71, Rayleigh number 2.5 10 and the maximum Mach number = 0.
of-ttsiibibtttioonn (OOD) fluid dynamics dataset designed for the development and benchmarking of machine learning models, particularly those that predict fluid dynamics under varying environmental conditions. This dataset includes simulations of tunnel and pool fires (representated as Prometheus-T and Prometheus-P in experiments), encom- passing wide range of fire dynamics scenarios modeled using fire dynamies simulators that solve the Havier-Stoker equations. Key features of the dataset include 25 different environmental settings with variations parameters such as Heat Release Rate (HRR) and ventilation speeds. In total, the Prometheus dataset encompasses 4.8 TB of raw data, which is compressed to 340 GB. It not only enhances the research on fluid dynamics modeling but also aids in the development of models capable of handling complex, real-world scenarios in safety-critical applications like fire safety management and emergency response planning.
Prometheus. Prometheus [48] is large-
WeatherBench. WeatherBench [36] is a benchmark dataset designed for the evaluation and compar- ison of machine learning models in the context of medium-range weather forecasting. is intended to facilitate the development of data-driven models that can improve weather prediction, particularly in the range from 14 days ahead. The dataset consists of historical weather data from multiple atmospheric variables, including temperature, pressure, humidity, wind speed, and geopotential height, at various global locations. The data is derived from the ERA5 reanalysis, which provides hourly estimates of the atmosphere's 's state at a resolution of 31 km for the period from 1950 to present. Its primary goal is to serve as a benchmarking tool compare the performance of machine learning-base based models against traditional numerical weather prediction methods. By focusing
21

---

data-driven techniques, it aims to push forward the development of models that can predict weather patterns in an interpretable and scalable manner, and thus contribute improving operational weather forecasting systems.
Kuroshio. Kuroshio [49] is a dataset designed to study the dynamics of the Kuroshio Current, a major oceanic current in the Pacific Ocean that flows from the east coast of Taiwan and Japan, obtained from the Copernicus Marine Environment Monitoring Service (CMEMS). Key features of the dataset include high temporal and spatial resolution measurements, covering daily and monthly intervals, which allow for in-depth studies on the seasonal and inter-annus variability of the Kuroshio Current. It also includes data on associated occanic phenomena such as eddy formation, upwelling. and interactions with surrounding currents like the T'ushima Current. This dataset covers the period from 1993 to 2024, and we use data from validation and testing.
for training, while data from 2021-2024 for
B Pseudocode of CoPS
Algorithm Algorithm workflow CoPS
Input: Initial observations Ug {uu(2, to), Met Query set Qset' Model parameters Oalii Hyperparameters Aorrr Àe.
2: Output: Predictions (i(lg, for (lg, xgg) E Quet-
3: /* Stage 1: Initial Encoding and Grid Mapping */
4: (t,(t))} +- Encodero.... (U)) P Encode observations to point embeddings D Map to initial latent grid state
Eq. 2, Eq.
5: =(o)) + GridMapper, map (th,(to)))
6: /* Stage 2: Iterative Latent Dynamics Prediction with Correction */
7: z + =(to)*; ty +- to: Ziras + {(to, ((to:((t))*))
8: while ty < max((,,))) (EQ... [ty) do
9: fnert_come +- to + Aoor
10: 2++1 ODESolve( Peoae 2x: [Kk, corr))
Continuous evolution
Eq. 7, 8, 11
11: Store evolution from 24. t0 2k+1 in Ztraj.
- 2641 + Ao Ae (zx) 12: *kt1
> Apply discrete correction
b Eq. 9, 10, 12
13: 2t - tx f- tneart_corr
14: end while
15: 1* Stage 3: Decoding to Query Locations */
16: Ý [}
for each query (ig) in Qset do Identify grid cell Veefl and its vertices {u,} for Zg.
18:
hyginit + GaborFilter(z.
Initialize query point representation
20:
hy.orid + MessagePass( (2 (z(tg)o, J, hygsese Eey tFe, }: 2 mmpp) Refine using grid states > Eq.
variant
21:
ū(tq, Fig) < MLLDecoder(1,,)) (hq.grid: O Odec)
22:
Add u(gg;"ng) to V.
23:
end for
24: A* Training: Parameters Oall = {Oenes Omaps Oode: Oac, Odec} are learned end-to-end by minimizing prediction error. return Y
C Details of Implementation
To ensure fairness, we conducted all experiments NVIDIA-A100 GPU using the MSE loss over 200 epochs. We used Adam optimizer with a learning rate of 10-3 for training. The batch size was set to 16.
22

---

C.1 Model hyper-paramees
The core architecture of the model consists of three main modules: the spatial information encoder, the multi-scale graph ODE module, and the neural auto-regressive correction module. The spatial information encoder employs Multiplicative Filter Networks to fuse partial observations with spatial coordinate information and encode them into eustomized geometric grids. For regular arranged datasets, the customized grid is set to the general resolution. For irregular arranged datasets, it is uniformly
128x128. The MFN is configured with layers and uses ReLU the activation function to ensure efficient feature extraction. The hidden dimension is set to 128. Then the multi- scale graph ODE module utilizes Runge-Kutta ode solver for numerical integration, with time step size of 0.25 to ensure accurate modeling of temporal continuity. Further, the neural auto- regresseeeee correction module performs corrections per integer time step. For this module, the Conv2d layer is downsampled to half the resolution, while the UpConv2d layer restores the grid to the original resolution. The parallel GroupConv2d operations are implemented with filter sizes of 3x3, 5x5, and 7x7. For inference, the correction weight À is set to 0.5 to balance correction strength and model stability. Finally, in the decoder, we use a single step Gabor filter transformation to initial the features of query coordinates, and perform a 2-layers message-passing update to obtain the corresponding predictions.
C.2 Baseline implementation
MAgNet [5]. We utilize the official implementation of MAgNet, utilizing a graph neural network variant of the model. The configuration involves five message-passingg steps. The architecture of all MLPs includes four layers, with each layer containing 128 neurons. Additionally, we set the dimensionality of the latent state at 128.
DINo [52]. We utilize the official implementation of DINo. Specifically, the encoder features an MLP, comprising three hidden layers with 512 neurons each, and Swish non-linearities. The dimension of each hidden layer is set to 100. Similarly, the dynamics function is realized through an MLP, which also includes three hidden layers, each containing 512 neurons and employing Swish activation function. The decoder is constructed with three layers, each with capacity of 64 channels.
ContiPDE [39]. ContiPDE formulates the task double observation problem. It utilizes recurrent GNNs to roll out predictions of anchor states from the IC, and employs spatio-temporal attention observer to estimate the state the query position from these anchor states. First, it utilize two- layered MLP with 128 neurons, with Swish activation functions to encode features form sparse observations. Further, it uses two-layered gated recurrent unit with hidden vector of size 128, and two-layered MLP with 128 neurons activated by the Swish function to realize the recurrent GNNs. Finally, employs multi-head attention mechanism to decode and utilizes multi-head attention mechanism to realize continuous query.
D Broader impacts
The CoPS framework has broad positive impacts in several areas. First, it provides crucial technical support scientific research, especially in geophysics, atmospheric science, and fluid dynamics. By performing continuous spatiotemporal simulations from sparse observational data, this method achieves high-preeision predictions with limited sensor numbers, significantly improving the accuracy and timeliness of weather forecasts and disaster warnings.
E
Limitations of This Study
While our proposed CoPS framework demonstrates significant advancements in modeling continuous spatiotemporal dynamics from partial observations, we acknowledge several limitations that provide avenues for future research. The first is the computational cost of multi-scale graph ODEs. It can be computationally intensive, especially for very fine-grained customized grids or large number of scales. Future work could explore more efficient graph neural network architectures or adaptive scaling mechanisms to mitigate this. The second is the assumption of Markov property in auto- correction. While this simplifies the model and makes it computationally tractable, real-world
23

---

physical systems might exhibit longer-rangg temporal dependencies that are not fully captured by this discrete correction mechanism.
F Proofs of Theorem 3.1
Let 6₁ = e(t denote the error just before the k-th correction (or at the start of the k-th ODE integration interval), and ex = =(tt denote the error just after the k-th correction. The initial error is co = e(to).
From assumption (b), after correction at time ty:
e Kek + dc.
(19)
From assumption (a), after ODE evolution from tx to tk+1:
cx+1 I E et + EODE-
(20)
We want to establish recurrence relation for ot. Substitute Eq(2) (with is replaced by le + 1 for ck+1) into Eq(1) (for et+1):
c S kE1+1 dc S nieExxtuure + EODE) + + Ke
Sboon ex + KEODE + 6c. (21)
Let Aeff ke Ateerr, Let KEODE + dc. Then the recurrence relation for the error after correction is:
c+t1 S aettee B.
(22)
This is linear first-order non-homogeneous recurrence relation. Unrolling this for K steps, starting from e
K S aame +B B
a + B) + B - + a.a B + B
(23)
K-1
Cef
K
B
Σ
off
/=0
Since Oef < 1, the geometric series sum is >KK-1 a
1-a
So,
1-afff
aell
S Cf
+
(24)
alef
This can be rewritten as:
B
Back
B
B
o S Ceff
K Colf
+
(25)
1 Oceff
1 Creff
1
Cteff
1
1-eeff
Ocff
The constant C2 defined in the theorem C2 floor for e Substituting C2 into Eq(7), we get:
kEou
This is the asymptotic error
1- 1ntt
I-
ox S ot (et - C2) +C2.
(26)
This matches the form of Equation (15) in the theorem statement, if we interpret [|e(t)] as ex (error after K corrections) and e(to) as et (error after the initial, possibly hypothetical, correction, which serves as the starting point for this recurrence). The number of correction steps is K [[t - to)/ Afcor).
Now for:
[(e(t)]| < C₁ - Offf
((t-to)/ Mearr) + C2.
(27)
From
o af (e) C2) Cz:
If C2 2 0, then we can set C1 = ex - C2. The bound becomes C10 K + Cz.
If 6/ C2 0, then et < Cz. Since a > O, the term a (et C2) is negative.
Thus, o < C2 a of (C2- et Cz. In this case, setting C1 0 ensures & a +C2 C2. which is true. So, C1 max(0. ot C2) correctly captures both cases and matches the definition in the theorem. The bound demonstrates that if ortf < 1. the term K + 0 as K - Consequently, the error exc converges towards the asymptotic error floor Cz.
24

---

G External Results
G.1 More Visualization Cases.
Out-horizan
CaPS
Comirut
5-75%
Figure 5: Qualitative comparison of our method and ContiPDE on the Navier-Stokes dataset under sparse initial observations (25% and 75%). In each panel, the first row displays the ground truth evolution, spanning both in-horizon (training) and out-horizon (extrapolation) timesteps. The second and third rows depict the predictions generated by our proposed CoPS and ContiPDE.
In horizon
Out bocizon
5-75%
CoPfs
Contiru
Figure 6: Qualitative comparison our method and ContiPDE on the WeatherBench dataset under sparse initial observations (25% and 75%). In each panel, the first row displays the ground truth evolution, spanning both in-horizon (training) and out-horizon (extrapolation) timesteps. The second and third rows depict the predictions generated by our proposed CoPS and ContiPDE.
To more clearly demonstrate the performance of our method, present detailed visualizations here. Figure and Figure 6 provide qualitative illustrations of CoPS's capabilities in modeling complex fluid dynamics from sparse initial data the Navier-Stokes dataset and the WeatherBench dataset, comparing its performance against ground truth and the ContiPDE baseline. We examine scenarios with initial conditions derived from both 25% and 75% of the full observations. The ground truth (first row in each panel) clearly shows the evolution of intricate flow patterns, including vortex formation and propagation, across both in-horizon and out-of-horizon time steps. When initialized with only 25% observations (top panel), our CaPS model (second row) successfully captures the essential dynamics, maintaining structural integrity and accurately predicting the advection vortices well into the extrapolation phase. In contrast, ContiPDE (third row), while capturing the general flow, struggles more with the sparsity, leading predictions that smoother and lose some of the details present in the ground truth, particularly in the out-of- of-oorizooooonooooooo predictions. This suggests CoPS's encoding mechanism and the interplay between its continuous ODE evolution and discrete auto- ceoo--orreetinn are more effective in inferring the complete state from limited information and robustly propagating Even with 75% initial data (bottom panel), where both models perform better, CoPS consistently exhibits a closer match to the ground truth's 's finer details and long-term behavior, highlighting its enhanced capacity for accurate and stable long-range forecasting continuous spatio-temporal domains.
25

---

G.2 Hyperparameter Analysis
We conduct sensitivity experiments with regard to correction hyperparameter (A) on Navier-Stokee and Prometheus datasets. To resolve your concern, we conduct experiments on both In-t and Ext-t settings with observation subsampling ratio of 50%. The results on Ext-t setting demonstrate the long-term prediction performance. The results are shown in Table 4, which indicate that neural auto-correction can indeed improve the performance our method, and the experimental results are robust to hyperparameter X.
Table 4: Hyperparameter sensitivity of À on the Navier-Stokes and Prometheus datasets with 50% subsampling ratio. We report MSE for In-t and Ext-t settings.
A
A= 0.1
À 0.2
À 0.5
À = 1.0
NAVIER- STOKES (IN-T)
3.244E-03
3.017E-03
2.925E-03
2.832E-03
2.964E-03
NAVIER-STOKES (ExI-1)
6,635E-03
6.172E-03
5.873E-03
5.764E-03
5.828E-03
PROMETHEUS (IN-T)
3.623E-03
3.542E-03
3.495E-03
3.374E-03
3.545E-03
PROMETHEUS (EX1-T)
7.016E-03
6.823E-03
6.747E-03
6.678E-03
6.837E-03
G.3 Noise Disturbance Robustness.
To evaluate the robustness of our model, we present the effects of observational noise on its per- formance and compare these results with those of other models. We quantify the noise using the channel-specific standard deviation tailored the dataset and have trained the model under various noise intensities (noise ratios set at 1%, 5%, 10%, 15%, and 20%). Experiments are conducted on and Pavier-Stokes and Prometheus datasets. Observations from Figure reveal that the proposed CoPS effectively maintains its performance with noise ratio below 5%, and demonstrates significant advantages over other baseline models when noise ratio increases over 10%. In contrast, we observe more pronounced performance degradation in the two interpolation-basee baseline models noise ratio raises over 10%, which indicates their weaker robustness against noise interference.
Navier-Stokes
Prometheus
0.007
MAghlt
0.012
Mighet
DINu
0.006
Ours
Ours
Test MSB
Test MSE
0.008
0.004
0.006
0.003
0.001
0.002
0.01
0.05
0.10
0.15
0.20
0.01
0.05
0.10
0.15
0.20
Noisc Rate
Noise Rate
Figure 7: Performance with regard to noise disturbance.
26


Reviews:
[{"review_id": "udVLObU1GX", "reviewer": "Reviewer_vtdp", "summary": "This study introduces CoPS, a data-driven framework designed to model continuous spatiotemporal dynamics in complex physical systems using partial observations. CoPS overcomes the discretization constraints of traditional methods by merging a multiplicative filter network to encode spatial coordinates with observation data, and multi-scale graph-based ordinary differential equations (ODEs) paired with a Markovian neural auto-correction module. Through experiments on diverse datasets—including Navier-Stokes fluid simulations and weather forecasting—the framework demonstrates enhanced performance in continuous space-time modeling and long-term prediction, particularly under sparse data conditions.", "questions": "Please provide responses to the above listed weakness. \n\nAlso, there are following additional questions:\n\n1. Should provide evaluations on efficiency of the proposed method and baselines, also the comparison of the number of parameters.\n2. How to decide the values of $\\mu$ and $\\gamma$ in Eq. (2)?\n3. How is relative position embedding designed?\n4. What is the effect of the hyperparameter $\\lambda$?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1.  CoPS proposes a data-driven approach combining multiplicative filter networks, customized geometric grids, and multi-scale graph ODEs to overcome discretization limitations in physics simulation, enabling spatio-temporal continuous prediction from partial observations. The framework integrates message-passing mechanisms, multi-scale graph ODEs for dynamic modeling, and a Markov-based neural auto-correction module for robust long-term predictions and error correction in nonlinear systems.\n\n2. Experiments on diverse datasets demonstrate state-of-the-art performance, especially in sparse data scenarios (25% observations) and long-term extrapolation beyond training horizons.\n\nWeaknesses:\n\n1. Core components (multiplicative filter network, multi-scale graph structure) resemble existing methods (MFN, GraphCast), lacking fundamental component-level innovations and relying on modular combinations.\n\n2. The inference cost is high since it involves an ODE simulation for a query point, which makes it expensive.\n\n3. The presentation of this paper is poor. In particular,  the Methodology Section is hard to follow. Some points need to be clarified:\n\n    (1) The structure and notations of the Methodology Section should be improved. For example, it is unclear how many models should be learned and how they are trained in this section.\n\n   \n    (2). What is the explicit formulation of the ODE in Eq. (7)? Why the continuous-time evolution could be modeled as such an ODE involving hierarchical graph structure G and hidden representations h?\n\n    (3). The mechanism of Neural Auto-correction is not very clear. Why it could help to reduce error accumulation? What is the relation between the module $r_\\phi$ and $E(\\cdot)$ and $D(\\cdot)$?", "was_revised": true, "final_justification": "My concerns are well clarified and addressed. I think the paper is now technically solid with comprehensive evaluation.\nThus, I raise my rating to \"4: borderline accept\". The reason why I do not provide a \"5: accept\" score for this paper is that I think its novelty is not strong enough.", "rebuttal": "Dear Reviewer vtdp, \n\nWe sincerely appreciate the time you’ve dedicated to reviewing our paper, as well as your valuable insights and support. Your positive feedback is highly motivating for us. Below, we address your primary concern and offer further clarification.\n\n> Q1. Core components (multiplicative filter network, multi-scale graph structure) resemble existing methods (MFN, GraphCast), lacking fundamental component-level innovations and relying on modular combinations.\n\nA1. Thanks for your valuable feedback. We agree that some of our core components are built upon powerful existing methods, and we have made sure to cite these foundational works. However, we wish to clarify that the novelty of our method lies not in its individual parts, but in the complete and functional system they form when synergistically integrated. \n\nSpecifically, we first employ a Gabor filter-based multiplicative filter network (MFN) to encode coordinate information into the initial state as a global frequency representation. Next, to address the challenge that grids in continuous modeling can be arbitrary and dynamic, we introduce a customized regular grid and design a dedicated message-passing mechanism to map features from the original unstructured domain onto it. \n\nSubsequently, for latent dynamics modeling on this regularized grid, we introduce the multi-scale graph ODE (MGO). While its hierarchical structure is inspired by GraphCast, we integrate it with an attention-based, multi-level message-passing scheme and uniquely combine it with a Neural ODE to learn both global and local dynamics in a continuous-time fashion. Furthermore, to mitigate the well-known issue of error accumulation in Neural ODEs, which arises from the difficulty of learning complex nonlinear features, we propose our neural auto-correction (NAC) module. This component performs adaptive corrections at discrete time steps.\n\nThe synergy of these modules is paramount to our model's success. Our extensive ablation study (Table 3) empirically validates this claim, showing that removing any of these components results in a significant degradation in performance. We will ensure this is articulated more clearly in the revised manuscript.\n\n> Q2. The inference cost is high since it involves an ODE simulation for a query point, which makes it expensive.\n\nA2. Thanks for your valuable comment regarding inference cost. We would like to clarify that the computationally intensive ODE simulation is performed only once for the entire system's latent state on a regularized grid, not on a per-query-point basis. Specifically, our method first performs a one-time encoding of the initial $N$ sensor observations onto a grid of size $G$ (complexity $\\mathcal{O}(N+G)$). The multi-scale Graph ODE then evolves the state of this entire grid over $S$ integration steps, which constitutes the primary computational cost of $\\mathcal{O}(S \\cdot G)$. Crucially, this cost is independent of the number of final query points, $M$. Finally, decoding the value at each of the $M$ query locations is a highly efficient decoder with a complexity of $\\mathcal{O}(M)$. Therefore, the total inference cost is $\\mathcal{O}(N+G + S \\cdot G + M)$, demonstrating that our framework amortizes the expensive dynamics simulation over an arbitrary number of spatial queries, with the cost scaling with temporal granularity ($S$) rather than spatial query density ($M$).\n\n> Q3. The structure and notations of the Methodology Section should be improved. For example, it is unclear how many models should be learned and how they are trained in this section.\n\nA3. Thanks for your comment. We wish to clarify that our proposed CoPS is a single, end-to-end trainable architecture, where all components are optimized jointly. As we have detailed in our response to the first question (A1), our framework is a synergistic system composed of several key modules (MFN, MGO, and NAC) that collectively achieve our goal. We acknowledge that we will carefully revise the structure and notations throughout the Methodology section in the revised manuscript. Furthermore, we will add a new subsection at the end of this section to provide a clear summary of the model implementation.\n\n> Q4. What is the explicit formulation of the ODE in Eq. (7)? Why the continuous-time evolution could be modeled as such an ODE involving hierarchical graph structure G and hidden representations h?\n\nA4. Thanks for your comment. (Eq. 7) instantiates a multi-scale graph ODE. The core of our methodological contribution here is the synergistic integration where we embed our multi-scale graph structure, along with its corresponding message-passing mechanism, directly into the Neural ODE framework to parameterize the derivative function $\\Phi$. This means that for any given state $h^t$, the function $\\Phi$ computes its instantaneous rate of change, $\\frac{dh^t}{dt}$, by performing a full multi-scale analysis as detailed in (Eq. 5) and (Eq. 6). Our rationale for this design is that many physical systems are governed by Partial Differential Equations (PDEs), where the temporal derivative of a field is determined by its current spatial configuration. The ODE framework provides a natural, continuous-time representation of this principle. \n\n> Q5. The mechanism of Neural Auto-correction is not very clear. Why it could help to reduce error accumulation? What is the relation between the module $r_\\psi$ and $E(\\cdot)$ and $D(\\cdot)$?\n\nA5. Thanks for your comment. To clarify the notation, the neural auto-correction module $r_\\psi$ in (Eq. 12) is indeed the entire composite function defined by the Encoder-Transition-Decoder structure, such that $r_\\psi(\\cdot) = \\mathcal{D}(\\mathcal{R}(E(\\cdot)))$. The purpose of this Neural Auto-correction (NAC) module is to mitigate the problem of error accumulation inherent in long-term predictions with Neural ODEs. While the ODE provides a continuous evolution path, it relies on a single learned vector field, where small approximation errors can compound over time, causing the solution to drift into physically implausible regions of the state space. Our NAC module acts as a corrector at discrete time intervals. At each correction step, the encoder $E(\\cdot)$ projects the state from the ODE into a more compact and stable latent space. Within this space, the transition block $\\mathcal{R}(\\cdot)$ applies a robust single-step mapping to learn latent correction. Finally, the decoder $D(\\cdot)$ projects the corrected state back to the original feature space. In essence, this mechanism does not attempt to perfect the ODE's learned vector field itself, but rather periodically re-initializes and constrains the solution trajectory to a learned manifold of valid physical states.\n\n> Q6. Should provide evaluations on efficiency of the proposed method and baselines, also the comparison of the number of parameters.\n\nA6. Thanks for your comment. To evaluate the efficiency our method and baselines, we have conducted experiments on the Navier-Stokes dataset. The results, presented in the table below, demonstrate that our method ranks second in raw efficiency metrics behind MAgNet, which employs nearest neighbor interpolation technique to generalize to new query points. The results clearly show that CoPS achieves its state-of-the-art predictive performance with exceptional efficiency. We will incorporate this complete analysis into our revised manuscript.\n\n|   |Param (MB)| Training time (h) | Inference time (s) |\n| - | - | - | - | \n| MAgNet|14.25|4.11| 3.25|\n| DINo|25.17 |6.74| 6.38|\n| ContiPDE|32.48|8.92|7.56|\n| Ours|22.53|5.27|5.51 |\n\n> Q7. How to decide the values of $\\mu$ and $\\gamma$ in Eq. (2)?\n\nA7. Thanks for your comment. Within the Gabor filter, the values for $\\mu$ and $\\gamma$ in our model are determined through a two-stage process: a principled initialization followed by end-to-end optimization, as they are treated as learnable parameters. Specifically, $\\mu$ is initialized uniformly at random within $[-1, 1)$ to ensure broad spatial coverage, while $\\gamma$ is sampled from a Gamma distribution, providing a robust, non-negative prior for the filter scale. Crucially, both are then wrapped as trainable weights. This means their final values are not manually set but are automatically learned and fine-tuned by the optimizer via backpropagation during training. This design transforms positional information into a feature representation rich with multi-scale frequency and orientation information, empowering the network to learn and reconstruct high-frequency details within the physical field, which is fundamental for modeling spatial continuity. The detailed descriptions will be included in the revised version.\n\n> Q8. How is relative position embedding designed?\n\nA8. Thanks for your comment. The role of the relative position embedding in our message-passing scheme (Eq. 4) is to inject a geometric inductive bias into the feature propagation process. The relative position embedding $\\phi(x_i, x_j)$ is generated by passing this difference vector through a single MLP, such that $\\phi(x_i, x_j) = \\text{MLP}(x_i - x_j)$. We will add this detail in the revised manuscript.\n\n> Q9. What is the effect of the hyperparameter λ?\n\nA9. Thanks for your comment. The effect of the hyperparameter λ is to balance the importance of Neural Auto-correction module in the phase of discrete correction. To dispel your concerns, we conduct sensitivity experiments on two datasets. The results indicate that this module can indeed improve the performance of our method, and the experimental results are robust to hyperparameter λ.\n\n| | λ=0|λ=0.1|λ=0.2|λ=0.5|λ=1.0|\n| - | - | - | - | - | - |\n|  Navier-Stokes(In-t)|3.244E-03|3.017E-03|2.925E-03|2.832E-03| 2.964E-03|\n|  Navier-Stokes(Ext-t)|6.635E-03|6.172E-03|5.873E-03 | 5.764E-03| 5.828E-03|\n|  Prometheus(In-t)|3.623E-03|3.542E-03 | 3.495E-03 | 3.374E-03| 3.545E-03 |\n|  Prometheus(Ext-t)|7.016E-03|6.823E-03|6.747E-03|6.678E-03| 6.837E-03 |"}, {"review_id": "jv1xW4oaib", "reviewer": "Reviewer_RVtr", "summary": "The authors propose a data-driven framework CoPS for the continuous modelling and prediction of dynamical systems. This allows for evolving systems in a grid-free and timestep-free way, constrained by sparse observations in time and space. The authors do this by combining several methodologies, namely multiplicative filter network for encoding, message passing for spatial modelling, and multi-scale graph ODEs for temporal modelling (in latent space). They show that CoPS outperforms a variety of benchmark on many continuous modelling for synthetic and real-world datasets.", "questions": "CoPS first performs an encoding to a customized but concrete grid. From the text in Sec. 3.1, it seems that points are mapped onto a uniform grid structure in the initial encoding stage. If this is the case, and there are sparse observations where points are sometimes very close together and sometimes very far apart, then this would require a large but very fine-grained uniform grid, leading to many many grid cells and thus be very expensive, and likely infeasible in a 3D setting. Could the authors please clarify? \n\nCould the authors provide more details on the baselines evaluation? Are the parameter counts for all the baselines roughly the same? How many graph scales are used for CoPS for the different tasks? What is the inference time for this method as compared to the baselines?", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The authors address discretization in both spatial and temporal dimensions, which allows the framework to be highly flexible. \n\nThe framework builds a collection of graphs on multiple scales, performs processing on each graph/scale independently, then mixes information between scales using an attention mechanism. \n\nThe authors use a neural correction module to address limitations when using numerical solvers. The mix of the standard solver to perform most of the work and the neural corrector to only model the residual is very nice.", "was_revised": true, "final_justification": "For both of my main concerns, the authors provided sufficient clarification. They also ran additional experiments and resolved my concerns about memory usage and scalability of the method.", "rebuttal": "Dear Reviewer RVtr, \n\nWe sincerely appreciate the time you’ve dedicated to reviewing our paper, as well as your valuable insights and support. Your positive feedback is highly motivating for us. Below, we address your primary concern and offer further clarification.\n\n> Q1. CoPS first performs an encoding to a customized but concrete grid. From the text in Sec. 3.1, it seems that points are mapped onto a uniform grid structure in the initial encoding stage. If this is the case, and there are sparse observations where points are sometimes very close together and sometimes very far apart, then this would require a large but very fine-grained uniform grid, leading to many many grid cells and thus be very expensive, and likely infeasible in a 3D setting. Could the authors please clarify?\n\nA1. Thanks for your valuable comment. This concern is entirely valid under the assumption that the grid resolution must be directly tied to the input data's spatial density. However, our approach is specifically designed to decouple the resolution of our customized latent grid from the spatial distribution of the sparse input observations, thereby avoiding this very issue. \n\nFirst, to clarify the mechanism: the customized grid in our framework acts as a structured, latent manifold whose resolution is a fixed hyperparameter, not a function of input point proximity. An observation point $p_i$ is mapped only to the vertices of the single grid cell that contains it. Even if ten points are tightly clustered within one cell, they all connect to the same few vertices. Information is then aggregated onto these vertices via a learnable message-passing scheme, which uses relative position embeddings to preserve the sub-cell spatial information. This design ensures that the primary computational cost is proportional to the chosen grid size (e.g., $H \\times W$), which remains constant and predictable, regardless of input point distribution.\n\nTo empirically validate this principle and directly address the insightful concern about 3D feasibility, we conducted new experiments on the 3D WeatherBench2 dataset [1]. We evaluated our model's performance and resource consumption across low, medium, and high grid resolutions. The results are presented below.\n\n|  | Low Res (32x64x60) | Medium Res (90x180x60) | High Res (180x360x60) |\n| :--- | :--- | :--- | :--- |\n| Total Grid Points | 123 K | 972 K | 3.89 M |\n| GPU Memory (VRAM) | 3.5 GB | 8.9 GB | 20.5 GB |\n| Inference Time (s) | 4.5 s | 19.0 s | 58.0 s |\n\nThe results reveals two crucial findings:\n1.  **Memory usage scales gracefully.** While the number of grid points grows by over 30 times from low to high resolution, the required VRAM increases by less than 6 times. This sub-linear growth is far from the prohibitive explosion one might fear and demonstrates the excellent memory efficiency of our approach in high-dimensional settings.\n2.  **Inference time shows strong sub-linear scaling.** Similarly, the inference time does not exhibit exponential growth but scales in a highly favorable, near-linear fashion relative to the number of grid points. This confirms that the computational cost is dictated by our manageable latent grid, not by complex interactions tied to input data geometry.\n\nIn all, both our model's design principle and these new 3D empirical results demonstrate that our method is indeed efficient and feasible for large-scale applications. The decoupling of the latent grid from the input data distribution is the key mechanism that ensures this scalability. We will include this experiment in our revised version.\n\n\nA2. Thanks for your valuable feedback. For the experiments presented in the paper, our CoPS model was configured with a consistent set of key hyperparameters to ensure robustness. A more detailed list of all model hyperparameters is available in the Appendix D.1 and D.3. Specifically, in the multi-scale Graph ODE (MGO) module, since the spatial resolutions of our experimental datasets are of a comparable order of magnitude, we adopted a consistent configuration of four graph scales across all our experiments.\n\nThen, to evaluate the efficiency our method and baselines, we have conducted experiments on the Navier-Stokes dataset. The results, presented in the table below, demonstrate that our method ranks second in raw efficiency metrics behind MAgNet, which employs nearest neighbor interpolation technique to generalize to new query points. The results clearly show that CoPS achieves its state-of-the-art predictive performance (as detailed in the paper)) with exceptional efficiency. This favorable balance underscores the effectiveness and practicality of our proposed architecture. We will incorporate this table into our revised manuscript.\n\n|   |Param| Training time | Inference time |\n| ------------ | ------ | ------   | ------ | \n| MAgNet    | 14.25 MB  | 4.11 h   | 3.25 s |\n| DINo      | 25.17 MB  | 6.74 h   | 6.38 s |\n| ContiPDE  | 32.48 MB  | 8.92 h   | 7.56 s |\n| Ours      | 22.53 MB  | 5.27 h   | 5.51 s |\n\n---\n[1] Rasp S, et al. WeatherBench 2: A benchmark for the next generation of data‐driven global weather models. Journal of Advances in Modeling Earth Systems 2024."}, {"review_id": "nF7GJLv0qU", "reviewer": "Reviewer_doai", "summary": "This paper introduces CoPS, a method for modeling physics simulations from partial spatial observations. The authors employ Gabor filters to generate representations and utilize Multiscale Neural Graph ODEs to model the dynamics. Additionally, they propose a Markovian correction term to mitigate cumulative errors in the ODE process. Experiments across various dynamical systems demonstrate improved performance in sparse observation, super-resolution, and temporal interpolation and extrapolation tasks.", "questions": "- What is the rationale behind using Gabor filters?\n- How is extrapolation/interpolation performed for arbitrary times when the correction is designed for discrete steps? Is the time interval $t_{k+1} - t_k$ always constant?\n- What is the effect of the correction hyperparameter?\n- Is Mean Squared Error (MSE) the most suitable metric for evaluating the impact of various components, or should other metrics like SSIM or PSNR be considered?", "limitations": "N/A", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- The model effectively handles multiple tasks, including interpolation, extrapolation, and super-resolution, with good results.\n- The authors provide theoretical justifications for the impact of latent corrections.\n- Despite its complexity, the paper is well-structured and easy to follow.\n\nWeakness:\n- While the work integrates numerous components for optimal results, many of these components are derived from previous studies, limiting the novelty of the approach.\n- The assumption that all dynamics can be learned from a single initial observation at time $t_0$ is questionable.\n- The phrase \"L arbitrary consecutive time points\" feels like an oxymoron.", "was_revised": true, "final_justification": "The authors' rebuttal clarified most of the questions I had about this work. Although, I still maintain that the work has limited novelty -- based on the results and discussion, I would recommend Weak Acceptance by increasing by score from 3 to 4.", "rebuttal": "Dear Reviewer doai, \n\nThank you for your valuable feedback on our manuscript. We have taken your comments seriously and have made the necessary revisions and additions to address the concerns raised. Below is our point-by-point rebuttal:\n\n> Q1. While the work integrates numerous components for optimal results, many of these components are derived from previous studies, limiting the novelty of the approach.\n\nA1. Thanks for your valuable comment. We agree that some of our core components are built upon powerful existing methods, and we have made sure to cite these foundational works. However, we wish to clarify that the novelty of our method lies not in its individual parts, but in the complete and functional system they form when synergistically integrated. \n\nSpecifically, we first employ a Gabor filter-based multiplicative filter network (MFN) to encode coordinate information into the initial state as a global frequency representation. Next, to address the challenge that grids in continuous modeling can be arbitrary and dynamic, we introduce a customized regular grid and design a dedicated message-passing mechanism to map features from the original unstructured domain onto it. \n\nSubsequently, for latent dynamics modeling on this regularized grid, we introduce the multi-scale graph ODE (MGO). While its hierarchical structure is inspired by GraphCast, we integrate it with an attention-based, multi-level message-passing scheme and uniquely combine it with a Neural ODE to learn both global and local dynamics in a continuous-time fashion. Furthermore, to mitigate the well-known issue of error accumulation in Neural ODEs, which arises from the difficulty of learning complex nonlinear features, we propose our neural auto-correction (NAC) module. This component performs adaptive corrections at discrete time steps.\n\nThe synergy of these modules is paramount to our model's success. Our extensive ablation study (Table 3) empirically validates this claim, showing that removing any of these components results in a significant degradation in performance. We will ensure this is articulated more clearly in the revised manuscript.\n\n> Q2. The assumption that all dynamics can be learned from a single initial observation at time $t_0$ is questionable.\n\nA2. Thanks for your comment. Our goal is not to learn from a single, isolated training example. Instead, our model is trained on a large dataset of **trajectories**, where each trajectory provides numerous examples of an initial state and its corresponding future states (e.g., pairs of $(u(t_0), u(t_1))$, $(u(t_0), u(t_2))$, etc.). By learning from thousands of such examples, the model's objective is to approximate a universal **evolution operator**, Φ, that governs how the system evolves from any valid initial state $u(t_0)$ to a future state $u(t_k)$. Therefore, at **inference time**, using a single observation at $t_0$ is the standard procedure to test the generalization and accuracy of the learned operator Φ.\n\nThis \"Initial Value Problem\" setup is not a novel invention of ours; rather, it is in direct alignment with the well-established paradigm of traditional numerical simulation that has been used for decades. There exist several Influential works like GraphCast [1] and Pangu-Weather [2]. The tremendous success of these models provides compelling evidence that for many high-dimensional physical systems, the state at a single point in time contains sufficient information to determine its short-to-medium-term evolution. Our work proudly follows this validated research framework.\n\n> Q3. The phrase \"L arbitrary consecutive time points\" feels like an oxymoron.\n\nA3. Thanks for your valuable feedback. To be precise: during the training stage, our model is indeed supervised using observations at a sequence of discrete time points from the dataset. The term 'arbitrary' was intended to describe the powerful capability of our model at the inference and evaluation stage. We recognize this distinction is critical and will state it with much greater precision in the revised manuscript to eliminate any ambiguity.\n\n\n> Q4. What is the rationale behind using Gabor filters?\n\nA4. Thank you for your insightful comment. The fundamental rationale for using Gabor filters is to overcome the well-documented spectral bias of standard MLP networks, which struggle to represent high-frequency functions and thus tend to produce overly smooth reconstructions of complex physical fields. To address this, our design employs Gabor filters to transform raw positional coordinates into a feature representation rich with multi-scale frequency and orientation information, directly empowering the network to learn and reconstruct the high-frequency details essential for physical fidelity.\n\n> Q5. How is extrapolation/interpolation performed for arbitrary times when the correction is designed for discrete steps? Is the time interval $t_{k+1} - t_{k}$ always constant?\n\nA5. Thanks for your comment. The task of predicting the system's state at any arbitrary time, whether for interpolation or extrapolation, is handled by our multi-scale Graph ODE module. It acts as the continuous dynamics engine. To predict a state at a query time $t_{query}$, we start from the most recent corrected state at a discrete time step $t_k$ and integrate the learned ODE forward from $t_k$ to $t_{query}$. This integration process is, by definition, continuous. The neural auto-correction (NAC) module, by contrast, acts as a stabilizer. It is invoked only at pre-defined, discrete time steps ($t_k, t_{k+1}$, etc.) to \"project\" the ODE's state back onto a learned manifold of physically plausible solutions, thereby preventing the long-term accumulation of numerical error. It does not participate in the prediction at arbitrary times between these steps. \n\nThen，for the time interval $t_{k+1} - t_{k}$, we did indeed use a constant correction interval, which typically corresponded to the interval between two consecutive frames in the training data (e.g., $\\Delta t = 1.0$). \n\n> Q6. What is the effect of the correction hyperparameter?\n\nA6. Thanks for your comment. We conduct sensitivity experiments w.r.t correction hyperparameter (λ) on Navier-Stokes and Prometheus datasets. To resolve your concern, we conduct experiments on both In-t and Ext-t settings with observation subsampling ratio of 50%. The results on Ext-t setting demonstrate the long-term prediction performance. The results are shown below, which indicate that neural auto-correction can indeed improve the performance of our method, and the experimental results are robust to hyperparameter λ.\n\n| | λ=0|λ=0.1|λ=0.2|λ=0.5|λ=1.0|\n| - | - | - | - | - | - |\n|  Navier-Stokes(In-t)|3.244E-03|3.017E-03|2.925E-03|2.832E-03| 2.964E-03|\n|  Navier-Stokes(Ext-t)|6.635E-03|6.172E-03|5.873E-03 | 5.764E-03| 5.828E-03|\n|  Prometheus(In-t)|3.623E-03|3.542E-03 | 3.495E-03 | 3.374E-03| 3.545E-03 |\n|  Prometheus(Ext-t)|7.016E-03|6.823E-03|6.747E-03|6.678E-03| 6.837E-03 |\n\n> Q7. Is Mean Squared Error (MSE) the most suitable metric for evaluating the impact of various components, or should other metrics like SSIM or PSNR be considered?\n\nA7. Thanks for your valuable comment. Our primary rationale for selecting Mean Squared Error (MSE) is its direct relevance to physical fidelity. In many scientific and engineering domains, MSE (or its square root, RMSE) is the standard and most direct metric for quantifying point-wise accuracy between a prediction and the ground truth, which aligns perfectly with our core objective of precise prediction. \n\nHowever, we fully agree that metrics like the SSIM and PSNR are indeed excellent for evaluating the perceptual and structural qualities. To provide a more holistic evaluation and address the concern, we have conducted experiments on the Navier-Stokes dataset (using 50% subsampling for training) to report these additional metrics. The results are summarized below.\n\n| Model | Task | MSE (↓) | SSIM (↑) | PSNR (↑) |\n| :--- | :--- | :--- | :--- | :--- |\n| MAgNet | In-t | 2.60E-02 | 0.826 | 25.85 |\n| | Ext-t | 4.29E-02 | 0.743 | 23.68 |\n| DINo | In-t | 1.07E-02 | 0.892 | 29.69 |\n| | Ext-t | 1.76E-02 | 0.834 | 27.54 |\n| ContiPDE | In-t | 8.34E-03 | 0.902 | 30.79 |\n| | Ext-t | 1.29E-02 | 0.873 | 28.89 |\n| **Ours (CoPS)** | **In-t** | **5.76E-03** | **0.942** | **33.40** |\n| | **Ext-t** | **9.82E-03** | **0.921** | **31.08** |\n\nAs the table demonstrates, our proposed method, CoPS, not only achieves superior performance on the MSE metric but also consistently outperforms all baseline models on both SSIM and PSNR. This confirms that our approach excels not only in point-wise accuracy but also in faithfully reconstructing the structural and fine-grained details of the underlying physical field. We will add this experiment in our revised manuscript. \n\n> Q8. Paper formatting concerns.\n\nA8. Thanks for your comment. We have performed a thorough proofread of the entire manuscript, correcting specific typographical errors (such as the instance of \"Marcov\" to \"Markov\") and making broader revisions to improve overall consistency, clarity, and presentation.\n\n---\n[1] Lam R, et al. Learning skillful medium-range global weather forecasting. Science, 2023.\n\n[2] Bi K, et al. Accurate medium-range global weather forecasting with 3D neural networks. Nature, 2023."}]

Author rebuttal:

Meta-review: The authors convincingly address concerns about scalability and efficiency: ODE integration is amortized over the latent grid (not per query), efficiency is competitive with prior work, and new 3D experiments demonstrate favorable memory/runtime scaling. Methodological clarity improved substantially through the discussion: the roles of ODE vs. NAC, the training objective (learning a universal evolution operator from many trajectories), hyperparameter handling (learned μ/γ, robust λ), and the mapping from irregular sensors to a fixed latent grid are now well articulated. The empirical study is broad and adds SSIM/PSNR alongside MSE, with CoPS consistently outperforming baselines in both pointwise and structural fidelity, particularly under sparse observation and long-horizon settings; data partitioning for both simulated (fixed Reynolds, varied initial conditions) and real datasets (ERA5 splits) is now clearly specified.

The primary weakness is limited component-level novelty: several modules build on known ideas, but the integration is coherent and greater than the sum of its parts, and the added analyses/ablation/sensitivity studies are insightful. Given the strong evidence of effectiveness, clarified methodology, scalability results in 3D, and positive reviewer updates (one accept, two borderline accepts after rebuttals), I judge that reasons to accept outweigh reasons to reject.
Decision: Accept (poster)