A Study on a Feedforward Neural Network to Solve Partial Differential Equations in Hyperbolic-Transport Problems

Abreu, Eduardo; Florindo, João Batista

doi:10.1007/978-3-030-77964-1_31

Cited by 9 publications

(17 citation statements)

References 21 publications

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“…In such cases, a vanishing viscosity approach or removing the triple point value using Rankine-Hugoniot condition seem to be straightforward solutions [54]. Similar strategies are even applied to PINNs in presence [18,19]. An extension of the proposed LPINN to higher spatial dimensions is possible using Radon transform [55].…”

Section: Discussionmentioning

confidence: 99%

“…The hyperparameters λ r , λ bc , and λ ic are scalars tuned to enhance the convergence. Augmenting the loss using more data points leads to faster convergence, especially in convection-dominated problems [19]. However, we intentionally refrain from using any data points to evaluate the convergence of PINNs as a solver (no-data regime).…”

Section: Traditional Pinnsmentioning

confidence: 99%

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Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Mojgani,

Balajewicz,

Hassanzadeh

2022

Preprint

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Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

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Section: Discussionmentioning

confidence: 99%

Section: Traditional Pinnsmentioning

confidence: 99%

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Mojgani,

Balajewicz,

Hassanzadeh

2022

Preprint

View full text Add to dashboard Cite

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“…Hyperbolic conservation law is used to simplify the Navier-Stokes equations in hemodynamics (Kissas et al, 2020). Hyperbolic partial differential equations are also addressed by Abreu and Florindo (2021): in particular, they study the inviscid nonlinear Burgers' equation and 1D Buckley-Leverett two-phase problem. They actually try to address problems of the following type:…”

Section: Hyperbolic Equationsmentioning

confidence: 99%

“…PINN methodology is also used also to address the 1D Buckley-Leverett two-phase problem used in petroleum engineering that has a non-convex flow function with one inflection point, making the problem quite complex (Abreu and Florindo, 2021). Results are compared with those obtained by the Lagrangian-Eulerian and Lax-Friedrichs schemes.…”

Section: Flows Problemsmentioning

confidence: 99%

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Cuomo¹,

Cola²,

Giampaolo³

et al. 2022

Preprint

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Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, and integral-differential equations. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages, the

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“…In real-world applications, these methods are also becoming more widely used. Eduardo et al in [13] applied deep learning methods to solve PDEs for transport models. Zichao et al in [14] proposed a new feed-forward deep network, called PDE-Net for estimating the dynamics of complex systems and their underlying implied PDE models.…”

Section: Introductionmentioning

confidence: 99%

Application of Lie Group-based Neural Network Method to Nonlinear Dynamical Systems

Ye¹,

Chaolu²

2022

Preprint

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A Study on a Feedforward Neural Network to Solve Partial Differential Equations in Hyperbolic-Transport Problems

Cited by 9 publications

References 21 publications

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Application of Lie Group-based Neural Network Method to Nonlinear Dynamical Systems

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