Many nonlinear Hamiltonian PDEs of interest are not integrable, yet exhibit long-time dynamics dominated by structures inherited from nearby integrable systems. This talk presents two complementary computational frameworks that address distinct problems arising in this setting.

The first framework, Sparse Identification of Lax Operators (SILO), addresses the question of integrability discovery. Given a dynamical system, SILO formulates the search for Lax operators as a sparse operator learning problem, identifying linear operators whose commutator structure is consistent with the observed evolution. This approach recovers known Lax pairs, discovers new ones, and provides a diagnostic for near-integrable structure even when exact integrability is broken.

The second framework, Proximal Spectral Coordinates (PROSPECTs), addresses a different problem: how to construct stable, interpretable reduced order models for nonlinear PDEs once a suitable operator structure is available. Rather than seeking compression through variance or global bases, PROSPECTs build reduced models from state-dependent spectral coordinates of a proximal Lax operator. Bound states capture coherent structures, while nearby continuum modes encode their deformation and radiation, yielding reduced models with strong long-time fidelity.

Although SILO and PROSPECTs target fundamentally different tasks, operator discovery versus model reduction, they form a natural pipeline. SILO identifies operator structure associated with integrability or near-integrability, while PROSPECTs use that structure to define dynamically meaningful coordinates for reduced modeling. Together, these methods suggest a unified operator-centric perspective on learning, structure, and reduction in nonlinear Hamiltonian dynamics.