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We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven…
Neural operators improve conventional neural networks by expanding their capabilities of functional mappings between different function spaces to solve partial differential equations (PDEs). One of the most notable methods is the Fourier…
Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods…
Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural…
In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable…
Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The Finite Element Method is often used as the numerical method of…
An accurate, physically-based, and differentiable model of soft robots can unlock downstream applications in optimal control. The Finite Element Method (FEM) is an expressive approach for modeling highly deformable structures such as…
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and…
Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep…
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically…
Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…
Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces…
The finite element method (FEM) is among the most commonly used numerical methods for solving engineering problems. Due to its computational cost, various ideas have been introduced to reduce computation times, such as domain decomposition,…
We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains…
Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical…
Correctly capturing intraoperative brain shift in image-guided neurosurgical procedures is a critical task for aligning preoperative data with intraoperative geometry for ensuring accurate surgical navigation. While the finite element…
Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…