Ziqian Li

Ziqian Li, Enrique Zuazua

We develop a reduced-order framework for optimizing mixing in two-dimensional incompressible flows. Instead of optimizing the full transport PDE, the method maximizes the length of advected material interfaces, leading to a finite-dimensional Hamiltonian control problem based on parametrized stream functions. We derive the continuous adjoint equations and reduced gradients, and discretize the forward and adjoint dynamics with the implicit midpoint rule. The resulting discrete adjoint is algebraically consistent with the derivative of the fully discrete objective, up to the tolerance of the nonlinear midpoint solves. The approach applies to bounded two-dimensional domains with smooth finite-dimensional stream-function parametrizations. Numerical experiments on cellular-flow and Doswell frontogenesis benchmarks show that the optimized time-dependent Hamiltonians generate near-exponential interface stretching and substantially faster decay of the $\dot{H}^{-1}$ mix-norm, in contrast with the polynomial behavior observed for stationary flows. When evaluated on a common reference transport solver, the interface-based controls produce faster $\dot{H}^{-1}$ decay than a Eulerian Sobolev-norm optimizer under a matched setup, while substantially reducing computational cost. We also identify a limitation of the reduced model: increasing the control basis may further improve the interface-length objective without yielding proportional gains in $\dot{H}^{-1}$ mixing, confirming that interface length is an effective but not fully faithful proxy for mixing in geometrically complex regimes.

Zheng Lu, Young Ju Lee, Jiwei Jia et al.

This paper develops a helicity-aware physics-informed neural network framework for incompressible non-Newtonian flow in rotational form. In addition to the energy law and the incompressibility constraint, helicity is a fundamental geometric quantity that characterizes the topology of vortex lines and plays an important role in the physical fidelity of long-time flow simulations. While helicity-preserving discretizations have been studied extensively in finite difference, finite element, and other structure-preserving settings, their realization within neural network solvers remains largely unexplored. Motivated by this gap, we propose a neural formulation in which vorticity is computed directly from the neural velocity field by automatic differentiation rather than learned as an independent output, thereby avoiding compatibility errors that pollute the helicity balance. To improve robustness and scalability, we combine two algorithmic ingredients: an overlapping spatial domain decomposition inspired by finite-basis physics-informed neural networks (FBPINNs), and a causal slab-wise temporal continuation strategy for long-time transient simulations. The local subnetworks are blended by explicitly normalized super-Gaussian window functions, which yield a smooth partition of unity, while the temporal evolution is advanced sequentially across time slabs by transferring the converged solution on one slab to the next. The resulting spatiotemporal framework provides a stable and physically meaningful approach for helicity-aware simulation of incompressible non-Newtonian flows.

Ziqian Li, Kang Liu, Yongcun Song et al.

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs and hence suffer from challenges in capturing temporal dynamics and generalization issues beyond training time frames. This paper introduces a deep neural ordinary differential equation (ODE) operator network framework, termed NODE-ONet, to alleviate these limitations. The framework adopts an encoder-decoder architecture comprising three core components: an encoder that spatially discretizes input functions, a neural ODE capturing latent temporal dynamics, and a decoder reconstructing solutions in physical spaces. Theoretically, error analysis for the encoder-decoder architecture is investigated. Computationally, we propose novel physics-encoded neural ODEs to incorporate PDE-specific physical properties. Such well-designed neural ODEs significantly reduce the framework's complexity while enhancing numerical efficiency, robustness, applicability, and generalization capacity. Numerical experiments on nonlinear diffusion-reaction and Navier-Stokes equations demonstrate high accuracy, computational efficiency, and prediction capabilities beyond training time frames. Additionally, the framework's flexibility to accommodate diverse encoders/decoders and its ability to generalize across related PDE families further underscore its potential as a scalable, physics-encoded tool for scientific machine learning.