Zhuoyuan Li

Zhuoyuan Li

Research Fellow, National University of Singapore

Posts by Collection

portfolio

publications

Learning to simulate partially known spatio-temporal dynamics with trainable difference operators

Published in arXiv, 2023

Recently, using neural networks to simulate spatio-temporal dynamics has received a lot of attention. However, most existing methods adopt pure data-driven black-box models, which have limited accuracy and interpretability. By combining trainable difference operators with black-box models, we propose a new hybrid architecture explicitly embedded with partial prior knowledge of the underlying PDEs named PDE-Net++. Furthermore, we introduce two distinct options called the trainable flipping difference layer (TFDL) and the trainable dynamic difference layer (TDDL) for the difference operators. Numerous numerical experiments have demonstrated that PDE-Net++ has superior prediction accuracy and better extrapolation performance than black-box models.

Recommended citation: Huang, X., Li, Z., Liu, H., Wang, Z., Zhou, H., Dong, B., & Hua, B. (2023). Learning to simulate partially known spatio-temporal dynamics with trainable difference operators. arXiv preprint arXiv:2307.14395. https://arxiv.org/pdf/2307.14395.pdf

Latent assimilation with implicit neural representations for unknown dynamics

Published in Journal of Computational Physics, 2024

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

Recommended citation: Li, Z., Dong, B., & Zhang, P. (2024). Latent assimilation with implicit neural representations for unknown dynamics. Journal of Computational Physics, page 112953. https://doi.org/10.1016/j.jcp.2024.112953

State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics

Published in Journal of Computational Physics, 2025

Data assimilation has become a crucial technique aiming to combine physical models with observational data to estimate state variables. Traditional assimilation algorithms often face challenges of high nonlinearity brought by both the physical and observational models. In this work, we propose a novel data-driven assimilation algorithm based on generative models to address such concerns. Our State-Observation Augmented Diffusion (SOAD) model is designed to handle nonlinear physical and observational models more effectively. The marginal posterior associated with SOAD has been derived and then proved to match the real posterior under mild assumptions, which shows theoretical superiority over previous score-based assimilation works. Experimental results also indicate that our SOAD model may offer improved accuracy over existing data-driven methods.

Recommended citation: Li, Z., Dong, B., & Zhang, P. (2025). State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics. Journal of Computational Physics, page 114240. https://doi.org/10.1016/j.jcp.2025.114240

Spend Wisely: Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping

Published in Advances in Neural Information Processing Systems, 2025

Modern foundation models often undergo iterative “bootstrapping” in their post-training phase: a model generates synthetic data, an external verifier filters out low-quality samples, and the high-quality subset is used for further fine-tuning. Over multiple iterations, the model’s performance improves—raising a crucial question: how should the total budget on generation and training be allocated across iterations to maximize final performance? In this work, we develop a theoretical framework to analyze budget allocation strategies. Specifically, we show that constant policies fail to converge with high probability, while increasing policies—particularly exponential growth policies—exhibit significant theoretical advantages. Experiments on image denoising with diffusion probabilistic models and math reasoning with large language models show that both exponential and polynomial growth policies consistently outperform constant approaches, with exponential policies often providing more stable performance.

Recommended citation: Yang, P., Feng, Y., Chen, Z., Wu, Y., & Li, Z. (2025). Spend Wisely: Maximizing Post-Training Gains in Iterative Synthetic Data Bootstrapping. in Advances in Neural Information Processing Systems, 38. https://neurips.cc/virtual/2025/loc/san-diego/poster/117235

In-context modeling as a retrain-free paradigm for foundation models in computational science

Published in arXiv, 2026

Building models that generalize across physical systems without retraining remains a central challenge in computational science. Here we introduce In-Context Modeling (ICM), a retrain-free paradigm that infers physical relationships directly from observational fields. Rather than encoding system-specific behavior in fixed parameters, ICM assimilates measurements as physical context and performs inference through a single forward pass. Trained in a physics-informed, label-free manner using governing equations, a single model generalizes across unseen materials, geometries, and loading conditions. Demonstrated on hyperelasticity, ICM integrates with finite-element simulations and is validated using experimental full-field measurements. Moreover, performance improves with increasing data diversity and computational budget, exhibiting favorable scaling behavior analogous to foundation models. By recasting physical modeling as in-context inference, this work establishes a transferable paradigm for retrain-free scientific learning and a foundation for scalable modeling across computational science.

Recommended citation: Li, L., Li, Z., Li, S., Zhan, K., Gao, H., Chen, C., & Yang, L. (2026). In-context modeling as a retrain-free paradigm for foundation models in computational science. arXiv preprint arXiv:2604.23098. https://arxiv.org/pdf/2604.23098

Hypothesis-driven construction of mesoscopic dynamics

Published in arXiv, 2026

Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. Data from each problem instance is then used to guide the identification of members within our hypothesis class, giving rise to accurate, robust and interpretable dynamical models. We empirically validate this framework on both data from continuum PDE models as a check, and on data arising from microscopic chain models for which exact meso-scale models are unknown. The proposed approach not only acts as an effective dynamics learner, but also offers vital interpretable diagnostics of the underlying physics.

Recommended citation: Li, Z., Zhu, A., & Li. Q. (2026). Hypothesis-driven construction of mesoscopic dynamics. arXiv preprint arXiv:2605.16211. https://arxiv.org/pdf/2605.16211

Ensemble Controlled-Flow Filtering for Implicit Data Assimilation

Published in arXiv, 2026

Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is defined as an energy tilt of the forecast distribution. We then propose the Ensemble Controlled-flow Filter (EnCF), which realizes this update through a stochastic controlled flow and learns the observation-dependent control by adjoint matching from terminal energy gradients. For simulator-defined observations, EnCF-LF learns a surrogate conditional energy from samples and applies the same controlled-flow solver. We prove ideal exactness, derive a one-step error decomposition, and establish non-accumulation of local errors under filter stability. Numerical results show that Kalman-type filters remain preferable for smooth additive-Gaussian observations, while the proposed methods are better suited to non-Gaussian, many-to-one, multimodal, and implicit observation models.

Recommended citation: Li, Z., Zhao, Y., & Li, M. (2026). Ensemble Controlled-Flow Filtering for Implicit Data Assimilation. arXiv preprint arXiv:2607.12975. https://arxiv.org/pdf/2607.12975

Fluorescent protein ticker tape (FPTT): Multiplexed recording of transcriptional dynamics in living cells and in vivo

Published in Science Advances, 2026

Recording and real-time imaging of promoter activities are critical for deciphering signaling cross-talk, but technologies for simultaneously capturing multiple transient events in living cells are lacking. Here, we design fluorescent protein-based ticker tapes (FPTT) for multiplexed, scalable, longitudinal recording of single-cell physiological activities by integrating multispectral monomeric fluorescent proteins with self-assembling protein fibers. FPTT logged dose-dependent, reversible endogenous cFos transcriptional histories in hippocampal neurons at 3-hour resolution over 8 days. We engineered FPTT variants for human nuclear factor κB (NF-κB), Janus kinase/signal transducer and activator of transcription 3 (STAT3), mechanistic target of rapamycin (mTOR), nuclear factor of activated T cells (NFAT), and adenosine 3′,5′-monophosphate (cAMP) signaling. This expanded toolset enabled quantification of cFos and NF-κB cross-talk in neurons, tracking of STAT3/cAMP dynamics during mouse liver injury, discovery of unexpected NFAT/STAT3 cross-talk, and characterization of cell cycle-dependent oscillating mTOR dynamics. Last, we achieved simultaneous analysis of four major pathways during T cell activation. FPTT provides a versatile platform to investigate transcriptional histories and signaling interplay, with broad applications in developmental biology and disease modeling.

Recommended citation: Wang, R., Jiang, J., Li, Z., Liu, T., Wang, Y., Xie, M., & Piatkevich, K. D. (2026). Fluorescent protein ticker tape (FPTT): Multiplexed recording of transcriptional dynamics in living cells and in vivo. Science Advances 12, eaef9406 https://www.science.org/doi/abs/10.1126/sciadv.aef9406

Entropy Production for Stationary Diffusions on Hilbert Spaces

Published in arXiv, 2026

We study entropy production for stationary diffusions on separable Hilbert spaces with possibly degenerate, state-dependent trace-class covariance. We first define the reversible–irreversible drift decomposition to provide a candidate drift for the reversed dynamics. Then by working directly with the invariant measure, we establish a lower bound in terms of the extended stationary Cameron–Martin energy of the irreversible drift, without requiring a diffusion representation of the stationary reversal. The bound implies infinite entropy production when the energy is infinite and we establish sufficient conditions for equality in the finite-energy regime. We also develop complementary criteria for identifying the stationary reversal as a Hilbert-space diffusion with constant or continuous state-dependent diffusion coefficients, respectively. Two nonlinear infinite-rank examples are provided to show that finite entropy production does not require the irreversible drift to lie in the covariance range, whereas entropy production may be infinite even with globally Lipschitz coefficients and injective covariance.

Recommended citation: Li, Z., Zhao, Y., & Zhu, A. (2026). Entropy Production for Stationary Diffusions on Hilbert Spaces. arXiv preprint arXiv:2610.08614. https://arxiv.org/pdf/2610.08614

talks

teaching

Teaching experience 1

Undergraduate course, University 1, Department, 2014

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Teaching experience 2

Workshop, University 1, Department, 2015

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