Yuanran Zhu

Yuanran Zhu, Jason M. Dominy, Daniele Venturi

We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and probability density function space formulations of the MZ equation. In particular, we derive errors bounds and sufficient convergence conditions for short-memory approximations, the $t$-model, and hierarchical (finite-memory) approximations. In addition, we derive computable upper bounds for the MZ memory integral, which allow us to estimate (a priori) the contribution of the MZ memory to the dynamics. Numerical examples demonstrating convergence of the proposed algorithms are presented for linear and nonlinear dynamical systems evolving from random initial states.

Yuanran Zhu, Daniele Venturi

We develop a new effective approximation of the Mori-Zwanzig equation based on operator series expansions of the orthogonal dynamics propagator. In particular, we study the Faber series, which yields asymptotically optimal approximations converging at least $R$-superlinearly with the polynomial order for linear dynamical systems. We provide a through theoretical analysis of the new method and present numerical applications to random wave propagation and harmonic chains of oscillators interacting on the Bethe lattice and on graphs with arbitrary topology.

Hardeep Bassi, Yuanran Zhu, Senwei Liang et al.

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to $O(n_T)$ from $O(n_T^2)$ if a $n_T$-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson's equation for quantum many-body systems.

Hardeep Bassi, Yuanran Zhu, Erika Ye et al.

We present RELift (Restrict, Evolve, Lift), a two-phase learning framework that couples coarse-grid numerical solvers with neural operators to super-resolve and forecast fine-grid dynamics for time-dependent partial differential equations (PDEs). In Phase 1, RELift learns a super-resolution operator that maps the solution on a coarse grid to a fine grid. In Phase 2, this learned operator is composed with a coarse-grid numerical integrator to construct an effective fine-grid propagator for the governing equation. We benchmark RELift on three canonical two-dimensional PDEs of increasing dynamical complexity -- the heat equation, the wave equation, and the incompressible Navier--Stokes equations -- and we further demonstrate its performance on a kinetic electron dynamics case study via the 1D1V Vlasov--Poisson system. Across all examples, RELift delivers high-fidelity super-resolution (Phase 1) and accurate long-horizon rollouts (Phase 2), outperforming standard super-resolution and neural operator baselines in both field-level error metrics and physics-relevant diagnostics. Finally, we provide error analysis of the effective fine-grid propagator, characterizing how approximation errors accumulate over time and explaining the observed numerical stability of the RELift framework.

Senwei Liang, Aditya N. Singh, Yuanran Zhu et al.

We propose a reinforcement learning based method to identify important configurations that connect reactant and product states along chemical reaction paths. By shooting multiple trajectories from these configurations, we can generate an ensemble of configurations that concentrate on the transition path ensemble. This configuration ensemble can be effectively employed in a neural network-based partial differential equation solver to obtain an approximation solution of a restricted Backward Kolmogorov equation, even when the dimension of the problem is very high. The resulting solution, known as the committor function, encodes mechanistic information for the reaction and can in turn be used to evaluate reaction rates.

Yuanran Zhu, Yu-Hang Tang, Changho Kim

We introduce a machine-learning framework named statistics-informed neural network (SINN) for learning stochastic dynamics from data. This new architecture was theoretically inspired by a universal approximation theorem for stochastic systems, which we introduce in this paper, and the projection-operator formalism for stochastic modeling. We devise mechanisms for training the neural network model to reproduce the correct \emph{statistical} behavior of a target stochastic process. Numerical simulation results demonstrate that a well-trained SINN can reliably approximate both Markovian and non-Markovian stochastic dynamics. We demonstrate the applicability of SINN to coarse-graining problems and the modeling of transition dynamics. Furthermore, we show that the obtained reduced-order model can be trained on temporally coarse-grained data and hence is well suited for rare-event simulations.

Yu-Hang Tang, Yuanran Zhu, Wibe A. de Jong

We present a simple algorithm for identifying and correcting real-valued noisy labels from a mixture of clean and corrupted sample points using Gaussian process regression. A heteroscedastic noise model is employed, in which additive Gaussian noise terms with independent variances are associated with each and all of the observed labels. Optimizing the noise model using maximum likelihood estimation leads to the containment of the GPR model's predictive error by the posterior standard deviation in leave-one-out cross-validation. A multiplicative update scheme is proposed for solving the maximum likelihood estimation problem under non-negative constraints. While we provide proof of convergence for certain special cases, the multiplicative scheme has empirically demonstrated monotonic convergence behavior in virtually all our numerical experiments. We show that the presented method can pinpoint corrupted sample points and lead to better regression models when trained on synthetic and real-world scientific data sets.