Reduced Operator Inference for Nonlinear Partial Differential Equations

Convergence of weak-SINDy Surrogate Models

“…Our results readily generalize to the multi-variable setting as many of the proof techniques rely on Hilbert space methods. As an application, we show how weak-SINDy techniques can be applied to POD system inference as considered in [17].…”

“…For a good overview of POD refer to [8,12,22]. Recently, as seen in [17], a non-intrusive method was devised to identify a system of ODEs over POD space. In [17], it was assumed that F has a representative system of ODEs that was at most quadratic in the temporal POD modes.…”

“…Recently, as seen in [17], a non-intrusive method was devised to identify a system of ODEs over POD space. In [17], it was assumed that F has a representative system of ODEs that was at most quadratic in the temporal POD modes. Here we exemplify that the weak-SINDy technique can act as a non-intrusive method to create a surrogate system of ODEs over POD space.…”

“…In particular, in the scalar ordinary differential equation (ODE) case, we show that (i) the surrogate dynamics converges towards the true dynamics and (ii) the solution of the surrogate model is reasonably close to the true solution, under the assumption of a bounded composition operator. In Section 4, this error analysis is extended to systems of ODEs, and the results are applied to identify a surrogate ODE system over coordinates defined by proper orthogonal decomposition (POD) for solutions to partial differential equations (PDEs), similar to the approach taken in [17]. Section 5 contains a variety of numerical examples exemplifying the main results of this paper.…”

Predictive Reduced Order Modeling of Chaotic Multi-scale Problems Using Adaptively Sampled Projections

“…Typically, the construction of projection-based ROMs includes two major steps: 1) An offline stage in which a small number of expensive FOM simulations are performed over a range of target parameters to obtain a low-dimensional subspace that approximates the high-dimensional FOM solution space; and 2) The online stage which projects the FOM equations onto the low-dimensional subspace, leading to a set of reduced equations for the ROM. Over the past decade, projected-based ROMs have been successfully demonstrated in many areas such as flow control [10,11,12], aeroelasticity [13,14], shape optimization [15,16,17,18], aerodynamics [19,20,21], building digital twins [22,23], hypersonics [24], rocket combustion [9,7,25,26], and detonation [27].…”