@article{anderson_farazmand_2024, title={Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks}, volume={498}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2023.112649}, abstractNote={We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anderson, William and Farazmand, Mohammad}, year={2024}, month={Feb} }
 @article{anderson_farazmand_2022, title={EVOLUTION OF NONLINEAR REDUCED-ORDER SOLUTIONS FOR PDEs WITH CONSERVED QUANTITIES}, volume={44}, ISSN={["1095-7197"]}, url={https://doi.org/10.1137/21M1415972}, DOI={10.1137/21M1415972}, abstractNote={Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time varying parameters have thus far been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. We demonstrate the efficacy of RONS on three examples: an advection-diffusion equation, the nonlinear Schrödinger equation and Euler’s equation for ideal fluids.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Anderson, William and Farazmand, Mohammad}, year={2022}, pages={A176–A197} }
 @article{anderson_farazmand_2022, title={Shape-morphing reduced-order models for nonlinear Schrodinger equations}, volume={4}, ISSN={["1573-269X"]}, url={https://doi.org/10.1007/s11071-022-07448-w}, DOI={10.1007/s11071-022-07448-w}, abstractNote={We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrödinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) the recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves, whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.}, number={4}, journal={NONLINEAR DYNAMICS}, publisher={Springer Science and Business Media LLC}, author={Anderson, William and Farazmand, Mohammad}, year={2022}, month={Apr} }