Title: Breaking the Discretization Barrier of Continuous Physics Simulation Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: dynamical system modeling
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}.


Peer reviews:
comments: 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.

STRENGTHS AND WEAKNESSES:
Strengths
- The model effectively handles multiple tasks, including interpolation, extrapolation, and super-resolution, with good results.
- The authors provide theoretical justifications for the impact of latent corrections.
- Despite its complexity, the paper is well-structured and easy to follow.

Weakness:
- While the work integrates numerous components for optimal results, many of these components are derived from previous studies, limiting the novelty of the approach.
- The assumption that all dynamics can be learned from a single initial observation at time $t_0$ is questionable.
- The phrase "L arbitrary consecutive time points" feels like an oxymoron.

QUESTIONS:
- What is the rationale behind using Gabor filters?
- 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?
- What is the effect of the correction hyperparameter?
- 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
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-10

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\nCoPS 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?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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 $D(\\cdot)$?\n\nQUESTIONS:\nPlease 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 $\\lambda$?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}