Applied Math
PhD Seminar

2026 Fall

The seminar of this semester is organized by Zihao Li and He Wang, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan. This section is partially sponsored by Shanghai Key Laboratory for Contemporary Applied Mathematics.

2026-11-05 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Gene expression is an inherently dynamic and stochastic process, and deciphering its underlying regulatory principles remains a central challenge in systems biology. Recent advances in single-cell and spatial transcriptomic technologies have generated unprecedented volumes of high-resolution data, offering new opportunities to reconstruct the dynamical landscapes of cellular states. In this review, we survey computational approaches for reconstructing stochastic dynamics from single-cell and spatial transcriptomic data, with a focus on methods grounded in optimal transport and stochastic differential equations. We discuss how these frameworks enable the inference of gene regulatory networks, the reconstruction of potential landscapes, and the quantitative characterization of cell fate transitions. We further highlight emerging AI-driven computational frameworks that integrate physical modeling with deep learning to address challenges such as data sparsity, noise, and spatial heterogeneity. Finally, we illustrate the application of these methods across a range of biological processes, including tumor progression and cancer heterogeneity, hematopoietic differentiation, and the cell cycle, and outline future directions for AI-enabled dynamical reconstruction in the era of spatial omics.

Past Presentations

2026-09-10 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This work considers stepsize schedules for gradient descent on smooth convex objectives. We extend the existing literature and propose a unified technique for constructing stepsizes with analytic bounds for an arbitrary number of iterations. This technique constructs new stepsize schedules by concatenating two stepsize schedules with fewer steps. Using this approach, we introduce two new families of stepsize schedules, achieving a convergence rate of $O(n^{-\log_2{(\sqrt 2+1)}})$ with state-of-the-art constants for the objective value and gradient norm of the last iterate, respectively. Furthermore, our analytically derived stepsize schedules either match or surpass the existing best numerically computed stepsize schedules.

2026-09-17 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Variational Monte Carlo (VMC) with neural network wavefunctions is a powerful approach to ground-state calculations, but its performance depends critically on wavefunction optimization. In this talk, we will introduce VMC and focus on geometry-aware optimization methods for neural network wavefunctions, in particular stochastic reconfiguration (SR) and its variants. Among them, SPRING, inspired by the randomized Kaczmarz method, has shown strong empirical performance but is sensitive to the choice of a momentum-like parameter $\mu$. We analyze this sensitivity and reveal a qualitative difference between $\mu<1$ and $\mu=1$. For $0\leq\mu<1$, we establish first-order convergence under weak moment assumptions allowing heavy-tailed Monte Carlo noise; for $\mu=1$, we construct counterexamples showing divergence through accumulation along kernel-related directions. Motivated by these results, we propose PRIME-SR, an adaptive SR method that adjusts the momentum parameter based on spectral information and subspace overlap. Numerical experiments show that PRIME-SR achieves performance comparable to well-tuned SPRING while substantially improving robustness in neural wavefunction optimization.

2026-09-24 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou--Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.

2026-10-08 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand PDE-constrained inverse problems arise throughout science and engineering, from medical imaging and fluid dynamics to geophysical exploration. A central example is full waveform inversion (FWI), which recovers subsurface velocity from seismic recordings by solving a severely ill-posed and nonconvex PDE-constrained optimization. Classical regularizers stabilize inversion but often fail to reproduce realistic geological structures, while supervised deep learning methods are efficient but generalize poorly to unseen structures, noise, and changing acquisition geometries. Diffusion models provide powerful data-driven priors by learning the score of Gaussian-corrupted marginals and generating samples via reverse-time SDEs or probability-flow ODEs. They have achieved strong results in image-domain inverse problems, but their use in PDE-governed problems remains challenging: existing diffusion-based algorithms introduce priors in various ways, yet physical constraints and the diffusion sampling process are not naturally consistent, and balancing prior guidance with stable, physics-aware optimization is difficult. This talk reviews PDE inverse problems, diffusion generative models, and diffusion-based inverse-problem algorithms, and then presents two recent works that embed pretrained diffusion priors into FWI. The first uses a decoupled latent-optimization formulation with an auxiliary physical variable, preserving classical smoothed-velocity initialization while letting the diffusion sampler provide a prior-consistent decoded velocity. The second couples a persistent physical state to the diffusion prior through a Gaussian bridge, mutually refining waveform fitting and denoiser-guided reverse sampling while separating wave-equation and denoiser gradients. Both methods are validated on OpenFWI and transferred to larger geological models under clean, noisy, and incomplete acquisitions. The results illustrate how diffusion priors can be made compatible with PDE constraints and provide practical algorithms for FWI and broader PDE-governed inverse problems.

2026-10-15 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We consider composite nonconvex optimization problems consisting of a smooth, possibly nonconvex function and a convex, possibly nonsmooth regularizer. To address the limitations of traditional proximal gradient methods, which typically require prior knowledge of the Lipschitz constant or line search for step size selection, we propose a simple adaptive proximal gradient method, termed AdaPGNC. The proposed method adaptively adjusts its step sizes using local upper and lower curvature estimates, allowing the step sizes to both increase and decrease. It is parameter-free, line-search-free, restart-free, and features a simple single-loop structure. Under standard assumptions, including Lipschitz continuity of the gradient, we establish an $\mathcal{O}(\varepsilon^{-2})$ iteration complexity for finding an $\varepsilon$-stationary point in the nonconvex setting, which is optimal in terms of $\varepsilon$. Our convergence analysis relies on a novel Lyapunov function that combines the objective function value gap with the squared norm of consecutive iterate differences. Furthermore, in the convex setting, the method requires no function value evaluations and achieves an $\mathcal{O}(1/k)$ convergence rate in objective function values, while linear convergence is established under strong convexity. Numerical experiments on neural network training, matrix completion, nonnegative matrix factorization, and logistic regression demonstrate the effectiveness and competitive performance of the proposed method.

2026-10-22 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Total least squares (TLS) extends least squares (LS) by accounting for errors in all variables, making it a natural framework for data fitting under measurement uncertainty. A better understanding of TLS can also inform the development of null-space estimation algorithms. Compared with LS, however, the theoretical foundations of TLS remain less fully developed. In this talk, we study TLS through the lens of backward error analysis. We derive an explicit expression for the optimal backward error of the general multidimensional TLS (mTLS) problem. To facilitate practical computation, we propose an efficiently computable estimate with a tight bound for the single right-hand-side case, together with a randomized approach for computing the backward error in the multidimensional setting. These results reveal close connections between LS and TLS. Finally, we present an algorithmic improvement to RQI-SPCGTLS, whose design also brings into focus the fundamental difficulties in extending LS theory to TLS.

2026-10-29 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This talk presents work on trust-region methods for smooth optimization. A new trust-region algorithm is developed that improves the theoretical performance of classical trust-region methods for convex problems. Furthermore, it is shown that acceleration techniques can be incorporated into trust-region-type methods.