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06/14/2023
Gauss Newton method for solving variational problems of PDEs with neural network discretizaitons
The numerical solution of differential equations using machine learning-...
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05/07/2023
Companion-Based Multi-Level Finite Element Method for Computing Multiple Solutions of Nonlinear Differential Equations
The use of nonlinear PDEs has led to significant advancements in various...
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04/06/2023
HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions
Due to the complex behavior arising from non-uniqueness, symmetry, and b...
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02/21/2023
Deep Learning via Neural Energy Descent
This paper proposes the Nerual Energy Descent (NED) via neural network e...
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10/13/2022
Bifurcation Analysis Reveals Solution Structures of Phase Field Models
Phase field method is playing an increasingly important role in understa...
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04/20/2022
Stability Preserving Data-driven Models With Latent Dynamics
In this paper, we introduce a data-driven modeling approach for dynamics...
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12/14/2021
Learn bifurcations of nonlinear parametric systems via equation-driven neural networks
Nonlinear parametric systems have been widely used in modeling nonlinear...
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11/02/2021
Two neural-network-based methods for solving obstacle problems
Two neural-network-based numerical schemes are proposed to solve the cla...
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09/01/2021
A Weight Initialization Based on the Linear Product Structure for Neural Networks
Weight initialization plays an important role in training neural network...
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07/24/2021
Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis
In this paper, we develop a sharp interface tumor growth model to study ...
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07/09/2021
An efficient greedy training algorithm for neural networks and applications in PDEs
Recently, neural networks have been widely applied for solving partial d...
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04/12/2021
A stochastic homotopy tracking algorithm for parametric systems of nonlinear equations
The homotopy continuation method has been widely used in solving paramet...
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02/25/2021
Power series expansion neural network
In this paper, we develop a new neural network family based on power ser...
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01/28/2021
Projection based model reduction for the immersed boundary method
Fluid-structure interactions are central to many bio-molecular processes...
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10/31/2020
Convergence analysis of neural networks for solving a free boundary system
Free boundary problems deal with systems of partial differential equatio...
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02/09/2020
An adaptive homotopy method for computing bifurcations of nonlinear parametric systems
In this paper, we present an adaptive step-size homotopy tracking method...
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12/06/2019
A randomized Newton's method for solving differential equations based on the neural network discretization
We develop a randomized Newton's method for solving differential equatio...
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10/03/2019