A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

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Computer-aided simulations are routinely used to predict a prototype's performance. High-fidelity physics-based simulators might be computationally expensive for design and optimization, spurring the development of cheap deep-learning surrogates. The resulting surrogates often struggle to generalize and predict novel scenarios beyond their training domain. We propose a two-stage methodology addressing the challenge of generalization. It employs physics-based simulators, supplemented with ordinary differential equations integrated into the recurrent architecture, to learn the intrinsic dynamics. The proposed approach captures the inherent causality and generalizes the dynamics irrespective of a data source. The presented numerical experiments encompass five fundamental structural engineering scenarios, including beams on Winkler foundations based on Euler-Bernoulli and Timoshenko theories, beams under moving loads, and catenary-pantograph interactions in railways. The proposed methodology outperforms conventional recurrent methods and remains invariant to data sources, showcasing its efficacy. Numerical experiments highlight its prospects for design optimization, predictive maintenance, and enhancing safety measures.

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Fracture distribution plays a significant role in the behavior of subsurface environments, affecting such activities as geothermal production, exploitation and management of groundwater resources, and long-term storage of nuclear waste and carbon dioxide. A key challenge in these and other applications is to estimate the fracture network properties from sparse and noisy observations. We evaluate the utility of cross-borehole thermal experiments for this task, using both physics-based particle-tracking (PBPT) heat-transfer approach and its deep neural network (DNN) surrogates. Synthetic data are provided by the PBPT simulations and used to train and test the DNN surrogates over a full range of the fracture network properties. We propose regionalized and step-by-step training techniques to reduce the computational cost of expensive PBPT forward solves over large ranges of the (to-be-estimated) parameters. Our numerical experiments suggest the feasibility of training a regionalized DNN surrogate over parameter ranges for which the PBPT solves are fast and extrapolating its predictions to parameter ranges with few additional data. We analyze the balance between computational cost and model accuracy, and provide both PBPT and DNN models for applications to others kinds of data.

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A two-dimensional particle-based heat transfer model is used to train a deep neural network. The latter provides a highly efficient surrogate that can be used in standard inversion methods, such as grid search algorithms. The resulting inversion strategy is utilized to infer statistical properties of fracture networks (fracture density and fractal dimension) from synthetic thermal experimental data. The (to-be-estimated) fracture density is well constrained by this method, whereas the fractal dimension is harder to determine and requires adding prior information on the fracture network connectivity. The method is tested on several fracture networks and hydraulic conditions.@en