Renormalized reduced models for singular PDEs

We present a novel way of constructing reduced models for systems of ordinary differential equations. In particular, the approach combines the concepts of renormalization and effective field theory developed in the context of high energy physics and the Mori‐Zwanzig formalism of irreversible statistical mechanics. The reduced models we construct depend on coefficients which measure the importance of the different terms appearing in the model and need to be estimated. The proposed approach allows the estimation of these coefficients on the fly by enforcing the equality of integral quantities of the solution as computed from the original system and the reduced model. In this way we are able to construct stable reduced models of higher order than was previously possible. The method is applied to the problem of computing reduced models for ordinary differential equation systems resulting from Fourier expansions of singular (or near-singular) time-dependent partial differential equations. Results for the 1D Burgers and the 3D incompressible Euler equations are used to illustrate the construction. Under suitable assumptions, one can calculate the higher order terms by a simple and efficient recursive algorithm. Spatial discretizations or Fourier expansions of the solutions of time-dependent partial differential equations (PDEs) lead to systems of ordinary differential equations (ODEs). The most difficult case arises when the solution of a PDE becomes singular in finite time. At such instants the solution of the PDE develops activity down to the zero length scale. A brute force numerical simulation (no matter how large) of such a solution is bound to fail because the simulation has a finite resolution and thus will be unable to resolve all the length scales down to the zero scale. When the solution develops activity at a scale smaller than the smallest scale available to the simulation, the numerically computed solution becomes underresolved. This leads to a rapid deterioration of the accuracy of the simulation. The notion of propagation of activity to smaller and smaller scales depends on the physical context of the PDE. In some cases, like the 3D Euler or Navier‐Stokes