Solving higher-order Lane-Emden-Fowler type equations using physics-informed neural networks: benchmark tests comparing soft and hard constraints

In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for solving different classes of singular ODEs, namely the well known second-order Lane-Emden equations, third order-order Emden-Fowler equations, and fourth-order Lane-Emden-Fowler equations. Two variants of PINNs technique are considered and compared. First, a minimization procedure is used to constrain the total loss function of the neural network, in which the equation residual is considered with some weight to form a physics-based loss and added to the training data loss that contains the initial/boundary conditions. Second, a specific choice of trial solutions ensuring these conditions as hard constraints is done in order to satisfy the differential equation, contrary to the first variant based on training data where the constraints appear as soft ones. Advantages and drawbacks of PINNs variants are highlighted

Modelling solar coronal magnetic fields with physics-informed neural networks

We present a novel numerical approach aiming at computing equilibria and dynamics structures of magnetized plasmas in coronal environments. A technique based on the use of neural networks that integrates the partial differential equations of the model, and called Physics-Informed Neural Networks (PINNs), is introduced. The functionality of PINNs is explored via calculation of different magnetohydrodynamic (MHD) equilibrium configurations, and also obtention of exact two-dimensional steady-state magnetic reconnection solutions (Craig & Henton 1995). Advantages and drawbacks of PINNs compared to traditional numerical codes are discussed in order to propose future improvements. Interestingly, PINNs is a meshfree method in which the obtained solution and associated different order derivatives are quasi-instantaneously generated at any point of the spatial domain. We believe that our results can help to pave the way for future developments of time dependent MHD codes based on PINNsComment: accepted in MNRA

Discutindo a educação ambiental no cotidiano escolar: desenvolvimento de projetos na escola formação inicial e continuada de professores

A presente pesquisa buscou discutir como a Educação Ambiental (EA) vem sendo trabalhada, no Ensino Fundamental e como os docentes desta escola compreendem e vem inserindo a EA no cotidiano escolar., em uma escola estadual do município de Tangará da Serra/MT, Brasil. Para tanto, realizou-se entrevistas com os professores que fazem parte de um projeto interdisciplinar de EA na escola pesquisada. Verificou-se que o projeto da escola não vem conseguindo alcançar os objetivos propostos por: desconhecimento do mesmo, pelos professores; formação deficiente dos professores, não entendimento da EA como processo de ensino-aprendizagem, falta de recursos didáticos, planejamento inadequado das atividades. A partir dessa constatação, procurou-se debater a impossibilidade de tratar do tema fora do trabalho interdisciplinar, bem como, e principalmente, a importância de um estudo mais aprofundado de EA, vinculando teoria e prática, tanto na formação docente, como em projetos escolares, a fim de fugir do tradicional vínculo “EA e ecologia, lixo e horta”.Facultad de Humanidades y Ciencias de la Educació

Solving differential equations using physics informed deep learning: a hand-on tutorial with benchmark tests

We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary

FINMHD: An Adaptive Finite-element Code for Magnetic Reconnection and Formation of Plasmoid Chains in Magnetohydrodynamics

International audienceSolving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small-scale magnetic islands called plasmoids. A new code is thus presented for the solution of two-dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end. A current–vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite-element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic-Galerkin scheme. The adaptivity of the code is illustrated on simplified test equations and finally for magnetic reconnection associated with the nonlinear development of the tilt instability between two repelling current channels. Varying the Lundquist number S has allowed us to study the transition between the steady-state Sweet–Parker reconnection regime (for S ≲ 104) and the plasmoid-dominated reconnection regime (for S ≳ 105). The implications for the understanding of the mechanism explaining the fast conversion of free magnetic energy in astrophysical environments such as the solar corona are briefly discussed

Approche numérique à l’usage du physicien pour résoudre les équations différentielles ordinaires.III. Cas d’équations différentielles aux dérivées ordinaires du second ordre en dimension 2. Application auxoscillateurs harmoniques/anharmoniques, et au mouvement Képlérien dans le plan

LicenceLe but de ce chapitre est d'étendre les méthodes numériques étudiées au chapitre précédent (Runge-Kutta et symplectiques, voir cel-01959896) à des systèmes en dimension 2. Le but final est l'intégration numérique de la trajectoire d'une particule dans un potentiel de type Képlérien (correspondant à la loi de la gravitation universelle de Newton). La propriété physique supplémentaire que les schémas numériques doivent essayer d'assurer pour les systèmes considérés ici, est la conservation du moment cinétique. Cette propriété est fondamentale (comme nous le verrons) pour la stabilité des orbites

Approche numérique à l'usage du physicien pour résoudre les équations différentielles ordinaires. II. Cas des oscillateurs harmonique/an-harmonique et du pendule non linéaire

LicenceDans ce chapitre, les méthodes numériques introduites au chapitre précédent (voir cel-01834234) sont généralisées pour résoudre des problèmes impliquant des équations différentielles aux dérivées ordinaires du second ordre. Pour illustrer l'effet des schémas numériques, nous choisissons d'abord l'oscillateur harmonique, le pendule Hamiltonien, et enfin un oscillateur an-harmonique. Il faut noter que nous introduirons une nouvelle méthode numérique dite symplectique, et dont les propriétés s'avèrent particulièrement intéressantes pour le type de problème exploré dans ce chapitre. Beaucoup de détails mathématiques sont fournis sur les comportements attendus des schémas pour le lecteur qui souhaite comprendre en profondeur. Cependant, il est aussi possible d'explorer de façon plus superficielle en 'jouant' tout simplement avec les programmes