Computation and Learning in High Dimensions (hybrid meeting)

The most challenging problems in science often involve the learning and accurate computation of high dimensional functions. High-dimensionality is a typical feature for a multitude of problems in various areas of science. The so-called curse of dimensionality typically negates the use of traditional numerical techniques for the solution of high-dimensional problems. Instead, novel theoretical and computational approaches need to be developed to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex problem-inherent structures and developing rigorous models to quantify the quality of information in a high-dimensional setting pose challenging tasks from both theoretical and numerical perspective. This has led to the emergence of several new computational methodologies, accounting for the fact that by now well understood methods drawing on spatial localization and mesh-refinement are in their original form no longer viable. Common to these approaches is the nonlinearity of the solution method. For certain problem classes, these methods have drastically advanced the frontiers of computability. The most visible of these new methods is deep learning. Although the use of deep neural networks has been extremely successful in certain application areas, their mathematical understanding is far from complete. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems

Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics

We present the neural-integrated meshfree (NIM) method, a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. NIM seamlessly integrates traditional physics-based meshfree discretization techniques with deep learning architectures. It employs a hybrid approximation scheme, NeuroPU, to effectively represent the solution by combining continuous DNN representations with partition of unity (PU) basis functions associated with the underlying spatial discretization. This neural-numerical hybridization not only enhances the solution representation through functional space decomposition but also reduces both the size of DNN model and the need for spatial gradient computations based on automatic differentiation, leading to a significant improvement in training efficiency. Under the NIM framework, we propose two truly meshfree solvers: the strong form-based NIM (S-NIM) and the local variational form-based NIM (V-NIM). In the S-NIM solver, the strong-form governing equation is directly considered in the loss function, while the V-NIM solver employs a local Petrov-Galerkin approach that allows the construction of variational residuals based on arbitrary overlapping subdomains. This ensures both the satisfaction of underlying physics and the preservation of meshfree property. We perform extensive numerical experiments on both stationary and transient benchmark problems to assess the effectiveness of the proposed NIM methods in terms of accuracy, scalability, generalizability, and convergence properties. Moreover, comparative analysis with other physics-informed machine learning methods demonstrates that NIM, especially V-NIM, significantly enhances both accuracy and efficiency in end-to-end predictive capabilities

Fracture analysis in continuously nonhomogeneous magneto-electro-elastic solids under a thermal load by the MLPG

AbstractA meshless method based on the local Petrov–Galerkin approach is proposed, to solve initial-boundary value problems of magneto-electro-elastic solids with continuously varying material properties. Stationary and transient thermal problems are considered in this paper. The mechanical 2-D fields are described by the equations of motion with an inertial term. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements, electric and magnetic potentials is approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time stepping method

Energy-Conserving Galerkin Approximations For Quasigeostrophic Dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [30] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) [32] for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finitedifference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom , and growth rates converge much more quickly with increasing for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.</p

A Streamline Upwind/ Petrov-Galerkin FEM based time-accurate solution of 3D time-domain Maxwell\u27s equations for dispersive materials

Although the simulation of most broadband frequency responses are made under the assumption of constant electromagnetic material parameters, this is not a valid assumption for many materials found in nature. In this dissertation the time-accurate solution of the 3D time-domain Maxwell’s equations for dispersive materials in a Streamline Upwind/Petrov-Galerkin framework is investigated. For this purpose the permittivity associated with a material is expressed as a matrix, enabling the solution of anisotropic material models with multiple poles. Here diagonally isotropic models with up to 2 poles are investigated. Near-field to far-field transformations are implemented to enable the solution of open boundary problems such as radiation patterns and radar cross sections. A unified perfectly matched layer absorbing boundary layer is implemented to efficiently terminate the computational region. Numerical simulations of these equations are tightly coupled together and compared against a loosely coupled approach to improve efficiency. An alternative diagonal stabilization matrix is proposed which is implemented and compared with a non-sparse stabilization matrix derived from the flux Jacobians. Along with this new stabilization parameter, scalability is improved by coupling the equations for the perfectly matched layer with those of Maxwell’s equations. Further efficiency gains are achieved by allowing for a variable number of equations to be solved throughout the domain