Physics-Informed Neural Networks for 2nd order ODEs with sharp gradients

In this work, four different methods based on Physics-Informed Neural Networks (PINNs) for solving Differential Equations (DE) are compared: Classic-PINN that makes use of Deep Neural Networks (DNNs) to approximate the DE solution;Deep-TFC improves the efficiency of classic-PINN by employing the constrained expression from the Theory of Functional Connections (TFC) so to analytically satisfy the DE constraints;PIELM that improves the accuracy of classic-PINN by employing a single-layer NN trained via Extreme Learning Machine (ELM) algorithm;X-TFC, which makes use of both constrained expression and ELM. The last has been recently introduced to solve challenging problems affected by discontinuity, learning solutions in cases where the other three methods fail. The four methods are compared by solving the boundary value problem arising from the 1D Steady-State Advection–Diffusion Equation for different values of the diffusion coefficient. The solutions of the DEs exhibit steep gradients as the value of the diffusion coefficient decreases, increasing the challenge of the problem

Mass Loss From Calving in Himalayan Proglacial Lakes

The formation and expansion of Himalayan glacial lakes has implications for glacier dynamics, mass balance and glacial lake outburst floods (GLOFs). Subaerial and subaqueous calving is an important component of glacier mass loss but they have been difficult to track due to spatiotemporal resolution limitations in remote sensing data and few field observations. In this study, we used near-daily 3 m resolution PlanetScope imagery in conjunction with an uncrewed aerial vehicle (UAV) survey to quantify calving events and derive an empirical area–volume relationship to estimate calved glacier volume from planimetric iceberg areas. A calving event at Thulagi Glacier in 2017 was observed by satellite from before and during the event to nearly complete melting of the icebergs, and was observed in situ midway through the melting period, thus giving insights into the melting processes. In situ measurements of Thulagi Lake’s surface and water column indicate that daytime sunlight absorption heats mainly just the top metre of water, but this heat is efficiently mixed downwards through the top tens of metres due to forced convection by wind-blown icebergs; this heat then is retained by the lake and is available to melt the icebergs. Using satellite data, we assess seasonal glacier velocities, lake thermal regime and glacier surface elevation change for Thulagi, Lower Barun and Lhotse Shar glaciers and their associated lakes. The data reveal widely varying trends, likely signifying divergent future evolution. Glacier velocities derived from 1960/70s declassified Corona satellite imagery revealed evidence of glacier deceleration for Thulagi and Lhotse Shar glaciers, but acceleration at Lower Barun Glacier following lake development. We used published modelled ice thickness data to show that upon reaching their maximum extents, Imja, Lower Barun and Thulagi lakes will contain, respectively, about 90 × 106 , 62 × 106 and 5 × 106 m3 of additional water compared to their 2018 volumes. Understanding lake–glacier interactions is essential to predict future glacier mass loss, lake formation and associated hazards

Physics-informed neural networks and functional interpolation for stiff chemical kinetics

This work presents a recently developed approach based on physics-informed neural networks (PINNs) for the solution of initial value problems (IVPs), focusing on stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). The framework developed by the authors combines PINNs with the theory of functional connections and extreme learning machines in the so-called extreme theory of functional connections (X-TFC). While regular PINN methodologies appear to fail in solving stiff systems of ODEs easily, we show how our method, with a single-layer neural network (NN) is efficient and robust to solve such challenging problems without using artifacts to reduce the stiffness of problems. The accuracy of X-TFC is tested against several state-of-the-art methods, showing its performance both in terms of computational time and accuracy. A rigorous upper bound on the generalization error of X-TFC frameworks in learning the solutions of IVPs for ODEs is provided here for the first time. A significant advantage of this framework is its flexibility to adapt to various problems with minimal changes in coding. Also, once the NN is trained, it gives us an analytical representation of the solution at any desired instant in time outside the initial discretization. Learning stiff ODEs opens up possibilities of using X-TFC in applications with large time ranges, such as chemical dynamics in energy conversion, nuclear dynamics systems, life sciences, and environmental engineering. © 2022 Author(s).12 month embargo; published online: 01 June 2022This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Pontryagin neural networks with functional interpolation for optimal intercept problems

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper

Pontryagin neural networks with functional interpolation for optimal intercept problems

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Physics-informed neural networks for rarefied-gas dynamics: Thermal creep flow in the Bhatnagar-Gross-Krook approximation

This work aims at accurately solve a thermal creep flow in a plane channel problem, as a class of rarefied-gas dynamics problems, using Physics-Informed Neural Networks (PINNs). We develop a particular PINN framework where the solution of the problem is represented by the Constrained Expressions (CE) prescribed by the recently introduced Theory of Functional Connections (TFC). CEs are represented by a sum of a free-function and a functional (e.g., function of functions) that analytically satisfies the problem constraints regardless to the choice of the free-function. The latter is represented by a shallow Neural Network (NN). Here, the resulting PINN-TFC approach is employed to solve the Boltzmann equation in the Bhatnagar-Gross-Krook approximation modeling the Thermal Creep Flow in a plane channel. We test three different types of shallow NNs, i.e., standard shallow NN, Chebyshev NN (ChNN), and Legendre NN (LeNN). For all the three cases the unknown solutions are computed via the extreme learning machine algorithm. We show that with all these networks we can achieve accurate solutions with a fast training time. In particular, with ChNN and LeNN we are able to match all the available benchmarks. © 2021 Author(s).12 month embargo; published online: 21 April 2021This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

CLASS OF OPTIMAL SPACE GUIDANCE PROBLEMS SOLVED VIA INDIRECT METHODS AND PHYSICS-INFORMED NEURAL NETWORKS

In this paper, a class of optimal space guidance control problems is solved using the combination of indirect method and Physics-Informed Neural Networks (PINNs). More specifically, we consider the class of optimal control problems with integral quadratic cost. The boundary value problems that arise from the application of the Pontryagin Minimum/Maximum Principle are solved via PINNs, which are particular neural networks where the training of the network is driven by the physics of the problem, modeled through Differential Equations. Three different PINN frameworks are considered, the standard PINN, the Physics-Informed Extreme Learning Machine (PIELM), and Physics-Informed Extreme Theory of Functional Connections (X-TFC). The main difference between standard PINN and PIELM with X-TFC is that with X-TFC initial and boundary conditions are analytically satisfied thanks to the so-called Constrained Expressions, introduced with the original Theory of Functional Connections (TFC). These expressions are a sum of a free-function, expanded as a single layer neural network trained via Extreme Learning Machine (ELM) algorithm, and a functional that analytically satisfies the boundary constraints. The results of this paper show the convenience of employing PINN frameworks to tackle this class of optimal control problems, especially PIELM and X-TFC, as they provide very good accuracy with low computational times