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Marimuthu Kalimuthu

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4 papers

2 author rows

ICLR Conference 2025 Conference Paper

Active Learning for Neural PDE Solvers

Daniel Musekamp
Marimuthu Kalimuthu
David Holzmüller
Makoto Takamoto
Mathias Niepert

Solving partial differential equations (PDEs) is a fundamental problem in engineering and science. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is costly to obtain. Active learning (AL) could help surrogate models reach the same accuracy with smaller training sets by querying classical solvers with more informative initial conditions and PDE parameters. While AL is more common in other domains, it has yet to be studied extensively for neural PDE solvers. To bridge this gap, we introduce AL4PDE, a modular and extensible active learning benchmark. It provides multiple parametric PDEs and state-of-the-art surrogate models for the solver-in-the-loop setting, enabling the evaluation of existing and the development of new AL methods for PDE solving. We use the benchmark to evaluate batch active learning algorithms such as uncertainty- and feature-based methods. We show that AL reduces the average error by up to 71\% compared to random sampling and significantly reduces worst-case errors. Moreover, AL generates similar datasets across repeated runs, with consistent distributions over the PDE parameters and initial conditions. The acquired datasets are reusable, providing benefits for surrogate models not involved in the data generation.

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TMLR Journal 2025 Journal Article

LOGLO-FNO: Efficient Learning of Local and Global Features in Fourier Neural Operators

Marimuthu Kalimuthu
David Holzmüller
Mathias Niepert

Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals due to swirling fluid motions caused by eddies and vortices. Faithfully modeling such signals using neural networks depends on the accurate reconstruction of moderate to high frequencies. However, it has been well known that deep neural nets exhibit the so-called spectral or frequency bias towards learning low-frequency components. Meanwhile, Fourier Neural Operators (FNOs) have emerged as a popular class of data-driven models in recent years for solving Partial Differential Equations (PDEs) and for surrogate modeling in general. Although impressive results have been achieved on several PDE benchmark problems, FNOs often perform poorly in learning non-dominant frequencies characterized by local features. This limitation stems from the spectral bias inherent in neural networks and the explicit exclusion of high-frequency modes in FNOs and their variants. Therefore, to mitigate these issues and improve FNO's spectral learning capabilities to represent a broad range of frequency components, we propose two key architectural enhancements: (i) a parallel branch performing local spectral convolutions and (ii) a high-frequency propagation module. Moreover, we propose a novel frequency-sensitive loss term based on radially binned spectral errors. This introduction of a parallel branch for local convolutions reduces the number of trainable parameters by up to 50% while achieving the accuracy of the baseline FNO that relies solely on global convolutions. Moreover, our findings demonstrate that the proposed model improves the stability over longer rollouts. Experiments on six challenging PDE problems in fluid mechanics, wave propagation, and biological pattern formation, and the qualitative and spectral analysis of predictions, show the effectiveness of our method over the state-of-the-art neural operator families of baselines.

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ICML Conference 2024 Conference Paper

Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations

Jan Hagnberger
Marimuthu Kalimuthu
Daniel Musekamp
Mathias Niepert

Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed, all of which suffer from the typical problems of Transformers, such as quadratic memory and time complexity. Furthermore, all prevalent architectures for PDE solving lack at least one of several desirable properties of an ideal surrogate model, such as (i) generalization to PDE parameters not seen during training, (ii) spatial and temporal zero-shot super-resolution, (iii) continuous temporal extrapolation, (iv) support for 1D, 2D, and 3D PDEs, and (v) efficient inference for longer temporal rollouts. To address these limitations, we propose Vectorized Conditional Neural Fields (VCNeFs), which represent the solution of time-dependent PDEs as neural fields. Contrary to prior methods, however, VCNeFs compute, for a set of multiple spatio-temporal query points, their solutions in parallel and model their dependencies through attention mechanisms. Moreover, VCNeF can condition the neural field on both the initial conditions and the parameters of the PDEs. An extensive set of experiments demonstrates that VCNeFs are competitive with and often outperform existing ML-based surrogate models.

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JAIR Journal 2021 Journal Article

Trends in Integration of Vision and Language Research: A Survey of Tasks, Datasets, and Methods

Aditya Mogadala
Marimuthu Kalimuthu
Dietrich Klakow

Interest in Artificial Intelligence (AI) and its applications has seen unprecedented growth in the last few years. This success can be partly attributed to the advancements made in the sub-fields of AI such as machine learning, computer vision, and natural language processing. Much of the growth in these fields has been made possible with deep learning, a sub-area of machine learning that uses artificial neural networks. This has created significant interest in the integration of vision and language. In this survey, we focus on ten prominent tasks that integrate language and vision by discussing their problem formulation, methods, existing datasets, evaluation measures, and compare the results obtained with corresponding state-of-the-art methods. Our efforts go beyond earlier surveys which are either task-specific or concentrate only on one type of visual content, i.e., image or video. Furthermore, we also provide some potential future directions in this field of research with an anticipation that this survey stimulates innovative thoughts and ideas to address the existing challenges and build new applications.

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