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This paper gives a theoretical framework for meshless latent dynamics models, which use neural fields conditioned on a latent state to reduce the order of time-dependent PDEs without a fixed grid. It proves a universal approximation result and Lipschitz stability bounds, and it claims comparable or better accuracy than mesh-based models with about 2% of the trainable parameters. Readers should note that the stability analysis points to an expressivity-stability trade-off, and the experiments cover only three benchmark PDEs.
Meshless Latent Dynamics Models for Model Order Reduction of time-dependent Partial Differential Equations
arXiv:2610.41043v1 Announce Type: cross Abstract: Latent dynamics learning has emerged as a prominent approach for Model Order Reduction (MOR) of time-dependent, parameterized Partial Differential Equations (PDEs). Traditional autoencoder-based approaches are intrinsically tied to fixed spatial discretizations during both training and inference, motivating the development of meshless methodologies. In this context, conditioning provides a natural mechanism for representing temporal evolution through spatially continuous functions, enabling mesh-agnostic Latent Dynamics Models (LDMs). In this work, we introduce a unified mathematical framework for mesh-agnostic LDMs based on conditioning in Neural Fields (NeFs). Leveraging a taxonomy of conditioning mechanisms (weak, strong, and pure), we formulate meshless LDMs as NeFs conditioned on a latent state. We establish a Universal Approximation Theorem (UAT) for this class of models and analyze their stability by deriving Lipschitz bounds for different conditioning mechanisms. Our analysis reveals an expressivity--stability trade-off, providing insight into why more expressive conditioning, despite its higher representational capacity, can potentially degrade performance during temporal extrapolation. Finally, we provide a comprehensive comparison of meshless and mesh-based architectures in the continuous limit of the spatial discretization, showing that meshless models guarantee grid-independent Lipschitz bounds, unlike mesh-based counterparts. Our experiments across Burgers' equation, Navier--Stokes flow past a cylinder, and linear advection show that meshless LDMs can achieve predictive accuracy comparable to or better than their mesh-based counterparts while using only 2% of the trainable parameters and exhibiting nearly constant error across spatial discretizations.
Abstract:Latent dynamics learning has emerged as a prominent approach for Model Order Reduction (MOR) of time-dependent, parameterized Partial Differential Equations (PDEs). Traditional autoencoder-based approaches are intrinsically tied to fixed spatial discretizations during both training and inference, motivating the development of meshless methodologies. In this context, conditioning provides a natural mechanism for representing temporal evolution through spatially continuous functions, enabling mesh-agnostic Latent Dynamics Models (LDMs). In this work, we introduce a unified mathematical framework for mesh-agnostic LDMs based on conditioning in Neural Fields (NeFs). Leveraging a taxonomy of conditioning mechanisms (weak, strong, and pure), we formulate meshless LDMs as NeFs conditioned on a latent state. We establish a Universal Approximation Theorem (UAT) for this class of models and analyze their stability by deriving Lipschitz bounds for different conditioning mechanisms. Our analysis reveals an expressivity--stability trade-off, providing insight into why more expressive conditioning, despite its higher representational capacity, can potentially degrade performance during temporal extrapolation. Finally, we provide a comprehensive comparison of meshless and mesh-based architectures in the continuous limit of the spatial discretization, showing that meshless models guarantee grid-independent Lipschitz bounds, unlike mesh-based counterparts. Our experiments across Burgers' equation, Navier--Stokes flow past a cylinder, and linear advection show that meshless LDMs can achieve predictive accuracy comparable to or better than their mesh-based counterparts while using only 2% of the trainable parameters and exhibiting nearly constant error across spatial discretizations.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2610.41043 [math.NA]
(or arXiv:2610.41043v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2610.41043 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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