| Abstract: |
| A general, data-informed, and theory-guided variational approach based on analytic parameterizations of unresolved variables will be presented to address the closure problem of stochastic systems. It relies on the Optimal Parameterizing Manifold (OPM) framework which allows, for deterministic chaotic systems away from the instability onset, to derive useful analytic formulas for such parameterizations.
These are obtained as homotopic deformations of parameterizations near criticality such as e.g. arising in center manifold reduction, and whose homotopy parameters are optimized away from criticality using data from the full model. Unlike other nonlinear-parameterization approaches such as those based on invariant/inertial or slow manifolds, the superiority of the OPM approach lies in its ability to alleviate the constraining spectral gap or timescale separation conditions.
In this talk, we present an extension of this program to stochastic partial differential equations (SPDEs) driven by additive noise, either white or of jump type. Analytic formulas of stochastic OPMs are derived. These parameterizations are optimized using a single solution path and are shown to represent efficiently the interactions between the noise and nonlinear terms in a given reduced state space, which are valid beyond the training solution path. Path-dependent non-Markovian coefficients depending on the noise history are shown to play a key role in these parameterizations especially when the noise is acting along the ``orthogonal direction`` of the reduced state space. Applications to stochastic transitions in SPDEs will be presented. |
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