Program

Abstract 

Artificial Intelligence Virtual Cell (AIVC) is increasingly emerging as a frontier in the interdisciplinary integration of biology and artificial intelligence. Its core vision is to construct digital twins capable of simulating and predicting the dynamic evolution of cellular states, thereby providing computational support for experimental design and mechanistic analysis (Cell 2024). Although omics foundation models inspired by Large Language Models (LLMs) have demonstrated potential in multi-task learning in recent years, recent evaluations indicate that they still face significant limitations in dynamic predictive capability and mechanistic interpretability, particularly when modeling complex biological processes (Nature Methods 2025). In this context, mathematical modeling methods that integrate biological priors and dynamic mechanisms are regaining attention (Cell 2025). However, the key data required for AIVC construction—single-cell multi-omics observations—commonly confront challenges such as sparse sampling and high-dimensional heterogeneity. As a result, traditional modeling approaches centered on differential equations encounter issues of insufficient observability and the curse of dimensionality in data-driven modeling, making it difficult to accurately depict high-resolution, continuous spatiotemporal dynamic processes.

To address these challenges, we have explored a unified modeling framework that integrates generative artificial intelligence methods and dynamical systems theory. By combining mathematical theories such as Optimal Transport, Schrödinger Bridge, and differential geometry with generative AI techniques like Flow Matching and diffusion models, this framework effectively infers continuous dynamic processes of complex state transitions—such as cell proliferation, apoptosis, differentiation, migration, and interactions—from static, heterogeneous single-cell omics temporal snapshots. Compared to black-box methods, this approach not only exhibits strong generative and generalization capabilities but also offers improved mechanistic interpretability. It enables the generation of cell state data across temporal scales and spatial structures, providing a promising direction for constructing dynamic virtual cell models with interpretability, predictive power, and the ability to integrate biological priors.

Abstract 

Data-driven AI often struggles with complex dynamical systems whose inputs are time-varying, discontinuous, or multiscale. I will introduce DePCoN, a mathematics-guided AI framework for robust parameter estimation. DePCoN replaces a difficult inverse problem with a hierarchy of smoothed surrogate problems and learns a continuation path from coarse to fine scales. By combining neural prediction with dynamical correction, DePCoN provides a concrete example of how mathematical structure can make scientific AI more stable, interpretable, and reliable.

Abstract 

Digital dentistry relies on cone-beam computed tomography (CBCT) for diagnosis and treatment planning. However, compact system design and low-dose requirements often lead to photon scattering and truncated projection data, making dental CBCT reconstruction an ill-posed inverse problem. These factors can produce shading artifacts and structural distortions in reconstructed images. In this presentation, we discuss several inverse problem formulations for dental CBCT under incomplete measurements. Projection-domain approaches use implicit neural representations (INRs) to model attenuation outside the reconstruction region and to compensate for missing measurements caused by field-of-view truncation. In addition, projection completion using an INR prior can be combined with analytic reconstruction methods such as FDK to obtain computationally efficient reconstructions. Image-domain approaches use generative adversarial networks (GANs) to reduce shading artifacts through learned image-to-image mappings. This work discusses different inverse problem formulations in digital dentistry, including both data-driven image restoration and physics-based reconstruction. These approaches address distinct sources of degradation in CBCT imaging and provide complementary improvements in structural consistency and image quality.

Abstract 

We study stochastic contextual bandits with heavy-tailed noise, where only a bounded (1 + ε)-th moment is assumed for some ε ∈ (0,1], under a broad class of mean reward functions. Existing heavy-tailed bandit algorithms have primarily focused on stochastic linear bandits, while the limited number of works beyond the linear setting either suffer from computational costs that scale poorly with the dimension d or incur per-round complexities growing with the time horizon t. We propose Hvt-SGD-TS, a novel algorithm that leverages the Huber loss-induced robustness and the computational efficiency of Online Gradient Descent and Thompson Sampling. Our algorithm updates parameter estimates in a phase-wise manner using the aggregated Huber loss, and does not require matrix inversion and expensive projection steps. We show that Hvt-SGD-TS achieves an overall computational cost of 𝒪(dT) for dimension d and time horizon T, shaving off a factor of d from the state-of-the-art algorithm, while attaining a regret bound of 𝒪̃(d¹·⁵T¹⁄⁽¹⁺ᵋ⁾) under mild assumptions on the distribution of contexts. This order is optimal in T, matching the lower bound for linear bandits. Experimental results demonstrate the effectiveness of our algorithm.

Abstract 

In this talk, I will introduce two data-driven frameworks developed to extract interpretable structure from complex biomedical data. The first, DiSPAH, addresses heterogeneity in longitudinal clinical datasets by separating latent disease progression pathways from individual-specific progression speed. This decomposition enables patients with similar symptoms but different temporal patterns to be distinguished and compared more precisely. The second, OrganoLink, focuses on multi-organ multi time-point single-cell data and aims to infer potential organ-to-organ influence relationships. By representing cellular state distributions geometrically and using optimal transport–based comparisons, OrganoLink identifies pairs of cell populations whose state changes may be linked. Together, these studies illustrate how data-driven quantitative modeling can transform complex biomedical measurements into interpretable insights into about disease progression and inter-organ coordination.

Abstract 

The success of deep neural networks relies heavily on the choice of learning rates, which govern the speed and stability of the training process. While classical line-search techniques provide principled mechanisms for selecting learning rates, their computational cost often makes them impractical for large-scale neural network optimization. In this talk, we present a relaxation-based framework that enables efficient determination of learning rates without the matrix–vector multiplications typically required by conventional line-search methods.

The proposed approach introduces two relaxed line-search formulations based on global and local Lipschitz estimates, leading to computationally efficient algorithms with rigorous theoretical guarantees. A key feature of the framework is a layerwise optimization strategy, in which network layers are updated sequentially rather than simultaneously. This perspective not only simplifies the mathematical analysis but also reveals an important characteristic of deep learning: different layers may require substantially different learning rates during training.

We discuss the derivation of the relaxed formulations, the associated convergence theory, and numerical experiments demonstrating their practical effectiveness. The results suggest that layerwise adaptation can provide a simple yet powerful alternative to conventional learning-rate selection strategies, offering new insights into the design of optimization algorithms for deep neural networks.

Abstract 

This talk explores recent theoretical and methodological advancements in deep generative models, especially, diffusion models. On the theoretical side, we demonstrate that diffusion models can circumvent the curse of dimensionality by discovering intrinsic low-dimensional structures within data distributions. We discuss this capability for both continuous and discrete variables and establish their theoretical optimality, showing that these advantages are driven by their feature learning abilities.

Methodologically, we introduce novel inference-time-adaptation/post-training techniques for diffusion models with discrete variables. Leveraging Doob's h-transform as our primary technical tool, we demonstrate that integrating this transform with the efficient sampling capabilities of diffusion models facilitates effective inference time adaptation without explicit parameter updates. Specifically, our approach enables reinforcement learning on the unmasking order distribution without requiring model updates, yielding substantial performance improvements, and we show the unmasking order adaptation has a significant impact on the performance improvement.

Abstract 

Artificial intelligence is becoming a powerful tool for scientific discovery, inverse design, and prediction of complex physical systems. In this talk, I will discuss how mathematical ideas shape AI for Science, focusing on generative models and foundation models. I will present examples from metamaterial discovery and weather prediction. In metamaterial discovery, generative and foundation-model approaches can be used for forward prediction and inverse design of structures with desired physical properties. In weather prediction, AI models learn complex spatiotemporal dynamics from large-scale data, while generative modeling provides a natural framework for uncertainty and extreme-event prediction. The talk will emphasize mathematical themes such as high-dimensional approximation, physical constraints, generative modeling, and reliable generalization, and will discuss open problems toward trustworthy scientific foundation models.

Abstract 

The phenomenological Landau-de Gennes (LdG) model is a powerful continuum theory to describe macroscopic liquid crystal (LC) phases. However, it is invariably less accurate and less physically informed than molecular-level models. We propose a neural network-based tensor (NN-tensor) model for LCs, supervised by an underlying molecular model. Our NN-tensor model not only attains energy precision comparable to the molecular model but also accurately captures the Isotropic-Nematic phase transition, which the LdG model cannot achieve. By embedding the NN-tensor model within a second neural network, we can efficiently compute stable LC configurations in a domain-free and mesh-free manner. We validate this approach with multiple examples for nematic LCs, demonstrating its ability to find physically relevant nematic configurations in diverse scenarios. We further apply the NN-tensor model to the more complex smectic LC phase. Strikingly, the NN-tensor model can quantitatively predict the smectic layer thickness and capture intricate microstructures such as Omega and T-shaped grain boundaries-features that current conventional approaches fail to resolve. These results demonstrate that the NN-tensor framework is a unified, efficient, and physically faithful route for computing rich LC configurations across multiple phases.

Abstract 

Machine learning has opened new possibilities for scientific computing and inverse problems. However, purely data-driven models often lack physical consistency and robustness, while mathematical models based on differential equations can be computationally expensive and difficult to construct for complex systems. Simulation-based machine learning seeks to bridge this gap by integrating data-driven learning with mathematical modeling and numerical simulation. By combining observation data, physical principles, and computational methods, it enables accurate prediction while preserving scientific interpretability.

In this talk, I will present recent advances in simulation-based machine learning, including surrogate models for large-scale physical simulations and physics-informed neural networks (PINNs) for crustal deformation analysis. Through applications to earthquake and tsunami modeling, I will discuss how machine learning and differential-equation-based modeling can complement each other and highlight emerging research directions in scientific machine learning. 

Abstract 

Estimating joint densities in high dimensions is a fundamental problem in statistics and machine learning. In many settings, the underlying dependence concentrates on lower-dimensional geometric structures that classical nonparametric estimators do not exploit, while purely parametric models impose overly rigid structure. In this work, we develop a structured nonparametric framework for joint density estimation based on an exponential-family ansatz in which the log-density decomposes into marginal potentials and an interaction potential capturing dependence beyond the marginals. Our framework admits a natural statistical-mechanics viewpoint that interprets maximum likelihood estimation as learning an energy landscape, which we use to unify three perspectives that have largely been studied separately. First, it provides a bridge to entropic optimal transport (EOT), where the interaction is fixed by a prescribed cost and the marginals by their empirical counterparts, whereas our formulation learns both from data. Second, jointly learning the marginal and interaction potentials disentangles marginal effects from intrinsic dependence, allowing the interaction component to be directly interpretable. Third, a one-sided constraint on the log-density induces a contact set that serves as a latent monotone-type scaffold, providing a positive-temperature relaxation of hard monotone-set constraints. Theoretically, we establish convergence rates for the MLE via a pseudo self-concordance bound on the third derivative of the free energy, which yields error bounds in terms of the complexity of the log-density function class. We empirically validate the proposed framework through numerical experiments.

Abstract 

Technological advances in spatial transcriptomics are critical for better understanding the structures and functions of tissues in biological research. The combination of intelligent or statistical algorithms and spatial transcriptomics has emerged to pave the way for deciphering tissue architecture. We have made great efforts to advance intelligent spatial transcriptomics and developed a group of STA- tools such as STAGATE, STAligner, STAMarker, STAGE, STASCAN, STALocaor, and STAMapper. In this talk, I will take these tools as examples to introduce how to utilize deep learning to model spatial transcriptomics data effectively. Moreover, I will shed light on some recent progress in spatial multi-omics integration with STAMO, directionally or temporally variable gene detection with STAVAG, and high-fidelity simulation and full-view 3D modeling of spatial transcriptomics with STADiffuser.

Abstract 

In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success across a wide range of fields. Operator learning has emerged as a powerful framework for parametric PDE problems: by learning the underlying solution operator, it enables rapid solution predictions as the input data (e.g., coefficients, sources, or boundary conditions) varies. In this talk, I will propose a new operator-learning method based on the classical finite element method , which is highly flexible across diverse settings without relying on any precomputed training data. Grounded in my recent results, the goal of this talk is to provide a broad mathematical investigation of the proposed method. First, through numerical experiments across a variety of scenarios, I will demonstrate practical effectiveness and versatility of the method. Second, leveraging mathematical theory for neural networks, I will develop a theoretical analysis of the approach and design theory-guided strategies that can substantially improve performance. Finally, from the perspectives of computational cost and memory usage, I will introduce a sparse-network-based approach inspired by the theory of finite element, and discuss its theoretical justification for simultaneously boosting computational efficiency, model accuracy, and memory efficiency.

Abstract 

We propose a topological data assimilation framework that integrates AI-assisted image analysis, topological data analysis, and mechanistic dynamical modeling to infer patient-specific pathophysiological states from skin eruption morphology. Clinical images are first processed to extract reproducible morphology-based features, and their spatial structures are encoded into topological descriptors capturing connectivity, heterogeneity, and lesion organization. These descriptors are then used to quantify similarity between observed eruption patterns and simulations generated by a mechanistic dynamical model of skin inflammation. 

As a proof of concept, we apply this framework to chronic spontaneous urticaria, where wheal morphology and treatment responsiveness vary among patients. Morphology-based virtual patient states inferred from image-derived and topological features reveal distinct dynamical regimes and predict morphology-dependent responses to in silico antihistamine treatment. These results suggest that visible skin morphology provides structural constraints for the otherwise ill-posed inverse problem of inferring hidden inflammatory dynamics. By integrating mechanistic mathematical modeling, AI-assisted image analysis, and topological data analysis, the proposed approach provides a mathematical framework for morphology-driven precision treatment in inflammatory skin diseases.

Abstract 

Causal discovery remains challenging in high-dimensional data, where combinatorial search over graphs becomes intractable. This talk presents two complementary works. First, we show that LLMs can reason about causal relationships from semantic variable descriptions and we integrate LLM-derived priors into data-driven discovery algorithms, letting observational evidence suppress false discoveries. Second, we present a functional diffusion model for causal ordering whose statistical estimates can be coupled with prior knowledges to control LLM-based causal reasoning without fine-tuning or prompt engineering.

Abstract 

This work presents a comprehensive framework combining PDE-based numerical methods and mesh-free deep learning approaches to stably and accurately approximate the viscosity solution of the Eikonal equation on complex three-dimensional domains. We first introduce a cell-centered finite volume method on polyhedral meshes within a vanishing-viscosity viewpoint, utilizing Soner boundary conditions to ensure correct solutions on non-convex geometries. This formulation achieves second-order experimental convergence for smooth test cases, scales efficiently in parallel computing, and significantly reduces computational costs far from source sets. Building upon this, we discuss three mesh-free deep learning solvers targeting the same viscosity solution. The first is a neural augmented Lagrangian method that leverages an implicit neural representation to formulate a constrained optimization problem, robustly enforcing Lipschitz gradient and Soner boundary constraints. The second is a viscosity-reduction variational approach for anisotropic Eikonal equations, which employs variable splitting and a normalized-output neural network architecture to ensure stable training on point clouds and discontinuous anisotropic metrics. Finally, we propose a transport-aware neural solver based on a Feynman-Kac stochastic representation, learning local gradient-aligned transport independent of the diffusion scale and naturally adapting to non-convex domains with obstacles.

Abstract 

We propose the Lower-envelope Eikonal Neural Operator (LENO), a physics-informed neural operator for eikonal equations with heterogeneous velocity fields and general source sets. LENO encodes the variational minimum principle of viscosity solutions by constructing source-wise candidate fields and taking their pointwise minimum. This lower-envelope structure enforces the source condition exactly and represents nonsmooth multi-source solutions without a soft boundary penalty. The model is trained without precomputed solution data, using a Godunov upwind residual as the sole training signal. Experiments on point sources and curve sources, sample-dependent piecewise velocity fields, and OpenFWI CurveVel-A and Style-A media show accurate agreement with fast-marching references. Additional validation shows that the Godunov residual outperforms forward and central finite-difference residuals on a high-resolution test, that velocity conditioning and the lower-envelope transform are essential in realistic heterogeneous media, and that the trained operator supports efficient batched evaluation and resolution transfer.

Abstract 

We extend the concepts of equicontinuity and distality from group actions to random group actions on metric spaces. Specifically, we prove that every equicontinuous random group action is pullback distal and that its pull-back orbit closures are minimal. Additionally, we introduce the Ellis semigroup theory for random group actions. Our results generalize established properties of equicontinuous and distal actions. Some short applications to processes, nonautonomous or stochastic differential equations are included.

Abstract 

Artificial intelligence is now transforming almost every area of science and technology. Despite its remarkable success, however, many modern AI models still rely heavily on empirical optimization strategies, and their fundamental mathematical principles remain only partially understood. This motivates the development of mathematical viewpoints that can explain, regularize, and guide data-driven learning systems. In this talk, I will discuss data science from the viewpoint of PDEs, with a particular focus on gradient flow type PDE dynamics. The main idea is to reinterpret several representative processes in AI and data science such as neural network training, data generation, and topological filtration, as evolution problems governed by energy, geometry, and diffusion. These examples suggest that PDEs can serve as a bridge between mathematical theory and modern data science. By viewing AI models as gradient flow like dynamical systems, we may gain new insight into their behavior, improve interpretability, and develop mathematically guided methods for future AI-driven scientific discovery.

Abstract 

While various data-driven discovery frameworks primarily focus on autonomous dynamics, environmental variations or external stimuli result in non-autonomous dynamics, which include time-varying parameters in their governing equations. The time-varying parameters make non-autonomous systems hard to analyze and predict only from measured discrete data. Here, we address this problem by proposing a data-driven framework that combines Koopman operator theory and artificial neural networks, named Koopman Identification of Non-autonomous Dynamics (KIND). KIND learns and predicts dynamics by embedding the original nonlinear and non-autonomous dynamics onto a Koopman space and further into a latent space. The learned model accurately predicts continuous trajectories from a single initial condition across both in-distribution and out-of-distribution time windows, thus providing a useful tool for predicting and understanding non-autonomous dynamics.