Plenary Talks

Variational Physics Informed Neural Networks:  quadrature rules, test functions, "a posteriori" error estimates and boundary conditions

Stefano Berrone, Politecnico di Torino

In this talk, we investigate the impact of different quadrature rules' precision and varying degrees of piecewise polynomial test functions on the convergence rate of Variational Physics Informed Neural Networks (VPINNs) when addressing elliptic boundary-value problems through mesh refinement. Employing a Petrov-Galerkin framework, we derive an a priori error estimate in the energy norm. The proposed interpolation operator is crucial to obtain an inf-sup stable method and to remove all spurious modes from the output of the neural network. Our findings may seem counterintuitive, suggesting that, for smooth solutions, the optimal approach for achieving a rapid error decay rate is to opt for test functions with the lowest polynomial degree while employing precision-oriented quadrature formulas.

We proceed by introducing an "a posteriori'' error estimator, comprising a residual component, a loss function term, and data oscillation terms. Our analysis demonstrates the reliability and efficiency of this estimator in controlling the energy norm error between the exact solution and the VPINN-derived solution.

Furthermore, a comprehensive exploration of four distinct approaches for enforcing Dirichlet boundary conditions in PINNs and VPINNs is performed. Traditionally, these conditions are enforced by introducing penalization terms in the loss function and carefully selecting corresponding scaling coefficients, a process that often demands resource-intensive tuning. Through a series of numerical tests, we establish that modifying the neural network's output to precisely align with the specified values yields more efficient and accurate solvers. Our most promising results emerge from the exact enforcement of Dirichlet boundary conditions, achieved by employing an approximate distance function. We also demonstrate that the variational imposition of these conditions using Nitsche's method yields suboptimal solvers.

(In-)stability in nonhomogeneous density fluids

Roberta Bianchini, IAC CNR

I will present a couple of recent rigorous results on the hydrostatic equations for nonhomogeneous density fluids, both in the positive and negative directions. Firstly, I will provide a proof of well-posedness with isopycnal diffusivity, establishing the validity of the hydrostatic (or shallow water) approximation. Secondly, in the fully inviscid and non-diffusive regime, I will provide an explicit steady state (which is unstable according to the Miles-Howard criterion) that allows to prove the breakdown of the hydrostatic limit and the generic ill-posedness of the limiting hydrostatic equations in finite regularity spaces.

Smooth Splines on Triangulations

Carla Manni, University of Rome "Tor Vergata"

Splines, in the classical sense of the term, are piecewise functions consisting of polynomial pieces glued together in a certain smooth way, usually by imposing equality of derivatives up to a given order. Besides their theoretical interest, splines find application in a wide range of contexts such as geometric modeling, signal processing, data analysis, visualization, and numerical simulation, just to mention a few. For many of these applications, a high smooth join between the different pieces is beneficial or even required. 

In the univariate case, splines of maximal smoothness, i.e., piecewise polynomials of degree p with C p - 1 joins, are probably the best known and most used splines. In fact, smoother splines give the same approximation order as less smooth splines of the same degree but involve fewer degrees of freedom and have less tendency to oscillate. When moving to the bivariate setting and considering polygonal partitions, e.g., triangulations, maximal smoothness is still very appealing but becomes an arduous task to achieve. To obtain splines of high smoothness on general triangulations in a stable manner, sufficiently large degrees have to be considered. An alternative is use lower-degree macro-elements that subdivide each triangle into a number of subtriangles (or more general subdomains). 

In this talk, we review the main issues concerning construction, and efficient representations in terms of proper bases, of highly smooth splines on triangulations, with a particular focus on a family of macro-elements of degree p and maximal smoothness p − 1. Afterwards, we briefly discuss some applications of smooth splines on (refined) triangulations in the context of Isogeometric Analysis. 

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Roberto Natalini, IAC,  CNR

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Maurizio Parton, University of Chieti-Pescara

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A general framework of implicit high-order schemes for hyperbolic systems 

Giuseppe Visconti, University of Rome "Sapienza"

This talk is concerned with the challenges of devising high-order implicit schemes for systems of hyperbolic conservation laws.

When solving hyperbolic systems, a source of difficulty is represented by stiff problems that occur when the speeds span different orders of magnitude. In this case, implicit schemes may become convenient because they require to choose a time step constrained by the CFL stability condition.

In contrast, implicit schemes are not constrained to the CFL condition and, thus, can be used to set up larger time step sizes. However, implicit methods may be more computationally expensive than explicit ones, since they require the solution of a system of equations, in general nonlinear, at each time step.

Here, we deal with efficient formulations of implicit high-order schemes. Achieving high-order accuracy requires to employ non-linear space limiters to prevent spurious oscillations. The use of such space-limiting procedures introduces a source of non-linearity which becomes computationally challenging when using implicit schemes.

The novel idea is to use a implicit first order schemes to pre-compute the non-linearities of the space-limiting procedure making the resulting implicit high-order scheme nonlinear just because of the non-linearity of the flux function. This approach is tailored to third order implicit schemes achieved by using a third order DIRK for the time integration and a third order space discretization performed either with CWENO or Discontinuous Galerkin approximations.