Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.

Physics-informed machine learning

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

Computer Methods in Applied Mechanics and Engineering

We present an adaptive algorithm for low Mach number reacting flows with complex chemistry. Our approach uses a form of the low Mach number equations that discretely conserves both mass and energy. The discretization methodology is based on a robust projection formulation that accommodates large density contrasts. The algorithm uses an operator-split treatment of stiff reaction terms and includes effects of differential diffusion. The basic computational approach is embedded in an adaptive projection framework that uses structured hierarchical grids with subcycling in time that preserves the discrete conservation properties of the underlying single-grid algorithm. We present numerical examples illustrating the performance of the method on both premixed and non-premixed flames.

Numerical simulation of laminar reacting flows with complex chemistry

Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains P.-O. Persson a J. Bonet band .J. Peraire c,* Department of Mathematics, University of California, Be'rkeley, Berkeley, CA 94720-3840, USA b School of Engineering, Swansea University, Swansea SA2 8PP, UK C Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-K utta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries. Key words: Discontinuous Galerkin, Deformable domains, Navier-Stokes, Arbitrary Lagrangian-Eulerian, Geometric Conservation 1. Introduction There is a growing interest in high-order methods for fluid problems, largely because of their ability to produce highly accurate solutions with minimum numerical dispersion. The Discontinuous Calerkin (DC) method produces stable discretizations of the convective operator for any order discretization, Moreover, it can be used with unstructured meshes of simplices, which appears to be a requirement for real-world complex geometries. In this paper, we present a high order DC formulation for computing high order solutions to problems with variable geometries. Time varying geometries appear in a number of practical applications such us rotor-stator flows, flapping flight or fluid-structure interactions, In such cases, it is necessary to properly account for the time variation of the solution domain if accurate solutions are to be obtained. For the Navier-Stokes equations, there has been a considerable effort in the development of Arbitrary Lagrangian Eulerian (ALE) methods to deal with * Corresponding author. Tel.: +1-617-25.3-1981: Fax.: +1-617-258-,514:3. Ematl address: peraire~mit .edu (J. Peraire). Preprint SUbJIlitted to Computer l\Jethods in Applied l\Jechanics and Engineering January 2009

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Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains - eScholarship

Learning constitutive relations using symmetric positive definite neural networks

Learning constitutive relations from indirect observations using deep neural networks

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

A family of position- and orientation-independent embedded boundary methods for viscous flow and fluid-structure interaction problems

A high fidelity multi-physics Eulerian computational framework is presented for the simulation of supersonic parachute inflation during Mars landing. Unlike previous investigations in this area, the framework takes into account an initial folding pattern of the parachute, the flow compressibility effect on the fabric material porosity, and the interactions between supersonic fluid flows and the suspension lines. Several adaptive mesh refinement (AMR)-enabled, large edge simulation (LES)-based, simulations of a full-size disk-gap-band (DGB) parachute inflating in the low-density, low-pressure, carbon dioxide (CO2) Martian atmosphere are reported. The comparison of the drag histories and the first peak forces between the simulation results and experimental data collected during the NASA Curiosity Rover's Mars atmospheric entry shows reasonable agreements. Furthermore, a rudimentary material failure analysis is performed to provide an estimate of the safety factor for the parachute decelerator system. The proposed framework demonstrates the potential of using Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI)-based simulation tools for future supersonic parachute design.

Daniel Z. Huang

Papers

Learning constitutive relations using symmetric positive definite neural networks

Learning constitutive relations from indirect observations using deep neural networks

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

A family of position- and orientation-independent embedded boundary methods for viscous flow and fluid-structure interaction problems

Modeling, Simulation and Validation of Supersonic Parachute Inflation Dynamics during Mars Landing