Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models

A difficult problem concerns the determination of magnetic field components within an experimentally inaccessible region when direct field measurements are not feasible. In this paper, we propose a new method of accessing magnetic field components using non-disruptive magnetic field measurements on a surface enclosing the experimental region. Magnetic field components in the experimental region are predicted by solving a set of partial differential equations (Ampere’s law and Gauss’ law for magnetism) numerically with the aid of physics-informed neural networks (PINNs). Prediction errors due to noisy magnetic field measurements and small number of magnetic field measurements are regularized by the physics information term in the loss function. We benchmark our model by comparing it with an older method. The new method we present will be of broad interest to experiments requiring precise determination of magnetic field components, such as searches for the neutron electric dipole moment.

mentioning

confidence: 99%

Magnetic field mapping of inaccessible regions using physics-informed neural networks

Coskun

Sel

Plaster

2022

“…To solve this problem, we directly enforce the initial and boundary conditions in the architecture of our PIANN. Thus, we follow the path of the second line in PINN literature 54 – 56 that enforces initial and boundary conditions as hard constraints. With respect to the first line, this provides several advantages.…”

Section: Methodsmentioning

confidence: 99%

“…The main drawback of this approach is, if this term was not exactly zero after training, the boundary condition is not completely satisfied. Other authors as Lagaris et al 54 or more recently, Schiassi et al 55 , 56 make use of the Extreme Theory of Functional Connections 57 to enforce the initial and boundary conditions in the solution. As in these works, we also enforce the initial and boundary conditions as hard constraints.…”

Section: Introductionmentioning

confidence: 99%

Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem

Rodriguez-Torrado

Ruiz²,

Cueto‐Felgueroso

et al. 2022

Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.

“…This challenge has been addressed by adaptive weighting strategies 8 – 11 , as well as theory of functional connections 12 , 13 . Despite these challenges, the effectiveness of the method has been demonstrated in a wide range of works, examples include turbulent flows 14 , heat transfer 15 , epidemiological compartmental models 16 or stiff chemical systems 17 .…”

Section: Introductionmentioning

confidence: 99%

Investigating molecular transport in the human brain from MRI with physics-informed neural networks

Zapf

Haubner

Kuchta

et al. 2022

In recent years, a plethora of methods combining neural networks and partial differential equations have been developed. A widely known example are physics-informed neural networks, which solve problems involving partial differential equations by training a neural network. We apply physics-informed neural networks and the finite element method to estimate the diffusion coefficient governing the long term spread of molecules in the human brain from magnetic resonance images. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small for accurate parameter recovery. To achieve this, we tune the weights and the norms used in the loss function and use residual based adaptive refinement of training points. We find that the diffusion coefficient estimated from magnetic resonance images with physics-informed neural networks becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented here are an important first step towards solving inverse problems on cohorts of patients in a semi-automated fashion with physics-informed neural networks.