# CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs

arXiv: https://arxiv.org/abs/2505.12944 · HTML: https://mlsanity.com/paper/2505.12944/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 May 2025.

## Built on by (most cited first, 8 of 8)

- [SMART: Scalable Mesh-free Aerodynamic Simulations from Raw Geometries using a Transformer-based Surrogate Model](https://mlsanity.com/paper/2601.18707/) (2026)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition](https://mlsanity.com/paper/2605.25115/) (2026)
- [STCO: Conditional Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2608.20477/) (2026)
- [HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries](https://mlsanity.com/paper/2609.18419/) (2026)
- [Cross-Domain Pretraining for Steady-State Neural CFD Surrogates](https://mlsanity.com/paper/2610.10398/) (2026)
- [Mesh-Native Tokenisation: Retrofitting Foundation Models to Unstructured Data](https://mlsanity.com/paper/2610.12618/) (2026)

## Abstract (arXiv)

Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics. However, performing these computations directly in the physical space often incurs significant computational costs. To address this issue, several neural surrogate models have been developed that operate in a compressed latent space to solve the PDE. While these approaches reduce computational complexity, they often use Transformer-based attention mechanisms to handle irregularly sampled domains, resulting in increased memory consumption. In contrast, convolutional neural networks allow memory-efficient encoding and decoding but are limited to regular discretizations. Motivated by these considerations, we propose CALM-PDE, a model class that efficiently solves arbitrarily discretized PDEs in a compressed latent space. We introduce a novel continuous convolution-based encoder-decoder architecture that uses an epsilon-neighborhood-constrained kernel and learns to apply the convolution operator to adaptive and optimized query points. We demonstrate the effectiveness of CALM-PDE on a diverse set of PDEs with both regularly and irregularly sampled spatial domains. CALM-PDE is competitive with or outperforms existing baseline methods while offering significant improvements in memory and inference time efficiency compared to Transformer-based methods.