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We introduce a novel continual learning method based on multifidelity deep neural networks. This method learns the correlation between the output of previously trained models and the desired output of the model on the current training…

Numerical Analysis · Mathematics 2024-07-01 Amanda Howard , Yucheng Fu , Panos Stinis

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

Numerical Analysis · Mathematics 2022-01-11 Yihao Hu , Tong Zhao , Shixin Xu , Zhiliang Xu , Lizhen Lin

We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an…

Machine Learning · Computer Science 2023-10-27 Valerii Iakovlev , Markus Heinonen , Harri Lähdesmäki

Continual learning remains a fundamental challenge in machine learning, requiring models to learn from a stream of tasks without forgetting previously acquired knowledge. A major obstacle in this setting is catastrophic forgetting, where…

Computation and Language · Computer Science 2025-12-18 Xiaodi Li , Dingcheng Li , Rujun Gao , Mahmoud Zamani , Feng Mi , Latifur Khan

Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…

Machine Learning · Computer Science 2023-03-31 Steven L. Brunton , J. Nathan Kutz

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be…

Machine Learning · Computer Science 2026-05-12 Koichi Taniguchi , Sho Sonoda

Continual learning is the problem of learning new tasks or knowledge while protecting old knowledge and ideally generalizing from old experience to learn new tasks faster. Neural networks trained by stochastic gradient descent often degrade…

Machine Learning · Computer Science 2019-11-27 David Rolnick , Arun Ahuja , Jonathan Schwarz , Timothy P. Lillicrap , Greg Wayne

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this…

Machine Learning · Computer Science 2026-03-20 Amartya Mukherjee , Maxwell Fitzsimmons , David C. Del Rey Fernández , Jun Liu

Continual learning refers to the ability of a biological or artificial system to seamlessly learn from continuous streams of information while preventing catastrophic forgetting, i.e., a condition in which new incoming information strongly…

Machine Learning · Computer Science 2019-07-04 German I. Parisi , Christopher Kanan

Catastrophic forgetting remains a fundamental challenge for neural networks when tasks are trained sequentially. In this work, we reformulate continual learning as a control problem where learning and preservation signals compete within…

Machine Learning · Computer Science 2026-02-05 Sander de Haan , Yassine Taoudi-Benchekroun , Pau Vilimelis Aceituno , Benjamin F. Grewe

Studies on generalization performance of machine learning algorithms under the scope of information theory suggest that compressed representations can guarantee good generalization, inspiring many compression-based regularization methods.…

Machine Learning · Computer Science 2019-10-16 Antoine Saporta , Yifu Chen , Michael Blot , Matthieu Cord

This paper addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior information, we propose a novel framework that learns a…

Statistics Theory · Mathematics 2026-02-09 Junxiong Jia , Deyu Meng , Zongben Xu , Fang Yao

Autonomous mobile robots require accurate human motion predictions to safely and efficiently navigate among pedestrians, whose behavior may adapt to environmental changes. This paper introduces a self-supervised continual learning framework…

Robotics · Computer Science 2022-02-16 Luzia Knoedler , Chadi Salmi , Hai Zhu , Bruno Brito , Javier Alonso-Mora

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

Machine Learning · Computer Science 2021-06-11 Sifan Wang , Paris Perdikaris

Using neural networks in practical settings would benefit from the ability of the networks to learn new tasks throughout their lifetimes without forgetting the previous tasks. This ability is limited in the current deep neural networks by a…

Machine Learning · Computer Science 2018-06-20 Risto Vuorio , Dong-Yeon Cho , Daejoong Kim , Jiwon Kim

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs), yet they often fail to achieve accurate convergence in the H1 norm, especially in the presence of boundary…

Numerical Analysis · Mathematics 2026-01-22 Qixuan Zhou , Chuqi Chen , Tao Luo , Yang Xiang

It has been observed that neural networks perform poorly when the data or tasks are presented sequentially. Unlike humans, neural networks suffer greatly from catastrophic forgetting, making it impossible to perform life-long learning. To…

Machine Learning · Computer Science 2023-01-31 Longhui Yu , Tianyang Hu , Lanqing Hong , Zhen Liu , Adrian Weller , Weiyang Liu

Continual learning is the ability to acquire new knowledge without forgetting the previously learned one, assuming no further access to past training data. Neural network approximators trained with gradient descent are known to fail in this…

Machine Learning · Computer Science 2021-11-05 Rodrigue Siry

Neural networks suffer from catastrophic forgetting and are unable to sequentially learn new tasks without guaranteed stationarity in data distribution. Continual learning could be achieved via replay -- by concurrently training externally…

Continual learning, or lifelong learning, is a formidable current challenge to machine learning. It requires the learner to solve a sequence of $k$ different learning tasks, one after the other, while retaining its aptitude for earlier…

Machine Learning · Computer Science 2022-04-25 Xi Chen , Christos Papadimitriou , Binghui Peng