Applied Math
PhD Seminar

2026 Spring

The seminar of this semester is organized by Jiamao Wu and Zhuangao He, 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.

2026-04-09 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (IBCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

2026-04-16 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Precise prediction of spatiotemporal dynamics over predictive horizons is constrained by the computational cost of high-fidelity solvers and the sparsity, noise, and irregularity of data. We introduce MERLIN, a Koopman-based framework that lifts dynamics to the evolution of learned observation functionals with near-linear progression, enabling full-field reconstruction at arbitrary resolutions. Theoretically, we develop a functional Koopman theory for PDEs and compensate for the loss of finite-dimensional linear invariance via the Mori–Zwanzig formalism, which augments the linear backbone with non-Markovian memory terms to improve predictive accuracy. Practically, MERLIN employs discretization-invariant function encoders that map partial, irregular observations to observables, and resolution-free function decoders that reconstruct states at arbitrary query points. Training under linear constraints yields an interpretable, low-dimensional model that captures principal modes, supports reduced-order modeling, and—augmented with memory correction—delivers stable long-horizon rollouts even in ultra-low-dimensional latent spaces.

Past Presentations

2026-03-05 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Currently, several finite difference methods are used to solve the Dirac equation in the semiclassical regime. However, finite difference methods have certain limitations in terms of accuracy and stability. The CNFD method, often employed to enhance stability, usually requires solving large linear systems, leading to significant computational costs. In this work, we conduct a rigorous analysis of the numerical error for the time-splitting methods applied to the Dirac equation in the semiclassical region. The dimensionless parameter $\epsilon \in (0,1]$, representing the reduced Planck constant, causes the solution field to exhibit rapid oscillations with characteristic wavelengths of order $O(\epsilon)$. We demonstrate that both $S_1$ and $S_2$ schemes preserve total mass conservation in a discrete sense. Furthermore, we establish error estimates for the time-splitting methods, clearly describing how the approximation error varies with time resolution $\tau$, spatial discretization step $h$, and the parameter $\epsilon$. We also derive error bounds for physical observables, and through numerical experiments, we validate the reliability of the error estimates.

2026-03-12 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we consider the numerical approximation of the Abels-Garcke-Grün model for a binary mixture of two viscous incompressible fluids with unmatched densities and viscosities. The system consists of a Navier-Stokes equation for the volume-averaged fluid velocity and a convective Cahn-Hilliard equation with Flory-Huggins potential for the phase-field variable. Based on the pressure stabilization method and an additional first-order stabilization term to the advective velocity in the Cahn-Hilliard equation, the scheme is fully decoupled and preserves the physical properties, i.e. the positivity-preserving property, which means that the discrete solution of the phase-field always stays in the physical interval (-1, 1) at a point-wise level, and the unconditional energy stability. Additionally, we perform a detailed optimal rate convergence analysis and derive error estimates. Numerical results are presented to validate the convergence rate and energy stability.

2026-03-19 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We study optimal learning rate schedules (LRSs) under the functional scaling law (FSL) framework, where the loss is written explicitly as a functional of the schedule. This formulation reveals a sharp phase transition governed by two exponents: a source exponent $s>0$ controlling signal learning and a capacity exponent $\beta>1$ controlling noise forgetting. In the easy-task regime, the optimal schedule follows a power decay to zero, with exponent determined by $\beta$. In the hard-task regime, the optimal schedule takes a warmup-stable-decay (WSD) form: it maintains the largest admissible learning rate for most of training and decays only near the end, with a vanishing decay fraction. We further analyze shape-fixed schedules, showing how the tail exponent governs both their optimality and their limitations through capacity saturation. This yields a principled evaluation of commonly used schedules such as cosine and linear decay. Finally, we apply the power-decay LRS to one-pass SGD for kernel regression and show that the last iterate attains the exact minimax-optimal rate.

2026-03-26 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand For a given matrix \( A \) in the real Schur form, the eigenvalue reordering algorithm updates \( A \) by an orthogonal matrix \( Q \) such that \( Q^{\top} AQ \) maintains the real Schur form while the eigenvalues of two adjacent diagonal blocks are swapped. By a series of updates, the diagonal blocks will eventually be arranged in a desired order. In LAPACK, this operation in double precision is performed by the routine DLAEXC, which is based on a direct swapping algorithm proposed by Bai and Demmel in 1993. In this work, we present a thorough error analysis of a modified direct swapping algorithm. By a carefully designed pivoting Givens QR factorization strategy, this algorithm can achieve an \( O(1) \cdot \mathbf{u} \Vert A \Vert \) error bound under quite mild assumptions, where \( \mathbf{u} \) is the machine precision. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.

2026-04-02 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This report introduces a machine learning method that combines Tensor Neural Networks (TNN) with homogenization theory for solving elliptic multiscale equations. The core advantage of TNN lies in its unique tensor structure, which allows the computation of high-dimensional neural network function integrals to be reduced to one-dimensional integrals. This enables the design of highly accurate high-dimensional integration methods, whose computational complexity scales only polynomially with the number of dimensions. Leveraging this feature, we design a high-precision solver for multiscale problems. Specifically, the original problem is first transformed via homogenization into a series of cell problems and a homogenized equation. These are then solved separately using TNN-based methods. Unlike conventional machine learning methods that rely on Monte Carlo sampling, our approach employs deterministic numerical integration, achieving high computational accuracy. In particular, for cases where the multiscale coefficients depend on both fast and slow variables, the corresponding cell problems are defined on high-dimensional domains; the TNN-based approach enables efficient and accurate computation for such cases compared to traditional methods, thereby extending the applicability of homogenization techniques. We also generalize this approach to elliptic multiscale eigenvalue problems.