Lie Generator Networks for Nonlinear Partial Differential Equations

When the system is linear, why should learning be nonlinear? Linear dynamical systems, the analytical backbone of control theory, signal processing and circuit analysis, have exact closed-form solutions via the state transition matrix. Yet when system parameters must be inferred from data, recent neural approaches offer flexibility at the cost of physical guarantees: Neural ODEs provide flexible trajectory approximation but may violate physical invariants, while energy preserving architectures do not natively represent dissipation essential to real-world systems. We introduce Lie Generator Networks (LGN), which learn a structured generator A and compute trajectories directly via matrix exponentiation. This shift from integration to exponentiation preserves structure by construction. By parameterizing A = S - D (skew-symmetric minus positive diagonal), stability and dissipation emerge from the underlying architecture and are not introduced during training via the loss function. LGN provides a unified framework for linear conservative, dissipative, and time-varying systems. On a 100-dimensional stable RLC ladder, standard derivative-based least-squares system identification can yield unstable eigenvalues. The unconstrained LGN yields stable but physically incorrect spectra, whereas LGN-SD recovers all 100 eigenvalues with over two orders of magnitude lower mean eigenvalue error than unconstrained alternatives. Critically, these eigenvalues reveal poles, natural frequencies, and damping ratios which are interpretable physics that black-box networks do not provide.

Identifying governing equations of nonlinear dynamical systems from data is challenging. While sparse identification of nonlinear dynamics (SINDy) and its extensions are widely used for system identification, operator-logarithm approaches use the logarithm to avoid time differentiation, enabling larger sampling intervals. However, they still suffer from the curse of dimensionality. Then, we propose a data-driven method to compute the Koopman generator in a low-rank tensor train (TT) format by taking logarithms of Koopman eigenvalues while preserving the TT format. Experiments on 4-dimensional Lotka-Volterra and 10-dimensional Lorenz-96 systems show accurate recovery of vector field coefficients and scalability to higher-dimensional systems.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.