This paper concerns the mathematical analyses of the diffusion model in machine learning. The drift term of the backward sampling process is represented as a conditional expectation involving the data distribution and the forward diffusion. The training process aims to find such a drift function by minimizing the mean-squared residue related to the conditional expectation. Using small-time approximations of the Green’s function of the forward diffusion, we show that the analytical mean drift function in DDPM and the score function in SGM asymptotically blow up in the final stages of the sampling process for singular data distributions such as those concentrated on lower-dimensional manifolds, and are therefore difficult to approximate by a network. To overcome this difficulty, we derive a new target function and associated loss, which remains bounded even for singular data distributions. We validate the theoretical findings with several numerical examples.

Stomata are important for plants as nodes linking the transpiration process and photosynthesis process. Accurate regulation of stomatal aperture balances water loss and carbon-dioxide uptake. In a recent work, a mathematical model is developed for seed-plant stomata, which exhibits rich dynamical behaviors and can semi-quantitatively explain responses of the stomata to environmental changes. In this work, we numerically study the dynamical behaviors of the stomatal model using phase diagrams with respect to a few important parameters. The phase diagrams are helpful for further validation of the model and experimental measurements of model parameters.

Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.

The matrix rank approximation has shown high effectiveness in the matrix rank minimization (MRM) problem, which aims to recover the underlying low-rank structure from the observed matrix by imposing the rank constraint. The nuclear norm, serving as a convex surrogate of matrix rank, is employed in the MRM problem by shrinking singular values of the observed entry. However, this substitution treats each singular value equally, which is virtually $ℓ_1$-norm penalty of the singular value vector. Theoretically, the rank function of the matrix can be considered as $ℓ_0$-norm of its singular values. Consequently, minimizing the nuclear norm frequently results in biased solutions in various applications. In this article, we first propose a novel nonconvex rank approximation, named tight and flexible rank (TFR) approximation, to describe rank function effectively. Specifically, the TFR approximation can more tightly approach the rank function and exhibit greater flexibility in handling diverse singular values, as compared to existing nonconvex rank approximations. Furthermore, we apply TFR approximation to matrix completion and develop a solving algorithm with guaranteed convergence based on the framework of proximal alternating minimization. Extensive experiments reveal that the proposed matrix completion model with TFR approximation outperforms several existing state-of-the-art convex and nonconvex methods.

A discontinuous Galerkin (DG) method on Bakhvalov-type (B-type) meshes for singularly perturbed Volterra integro-differential equations (SPVIDEs) is proposed. We derive abstract error bounds of the DG method for the SPVIDEs in the $L^2$-norm. It is shown that the approximate solution generated by the DG method on B-type meshes has optimal convergence rate $k + 1$ in the $L^2$-norm, when using the piecewise polynomial space of degree $k.$ Numerical simulations demonstrate the validity of the theoretical results.

We propose the optimal production transport model, which is an extension of the classical optimal transport model. We observe in economics, the production of the factories can always be adjusted within a certain range, while the classical optimal transport does not take this situation into account. Therefore, differing from the classical optimal transport, one of the marginals is allowed to vary within a certain range in our proposed model. To address this, we introduce a multiple relaxation optimal production transport model and propose the generalized alternating Sinkhorn algorithms, inspired by the Sinkhorn algorithm and the double regularization method. By incorporating multiple relaxation variables and multiple regularization terms, the inequality and capacity constraints in the optimal production transport model are naturally satisfied. Alternating iteration algorithms are derived based on the duality of the regularized model. We also provide a theoretical analysis to guarantee the convergence of our proposed algorithms. Numerical results indicate significant advantages in terms of accuracy and efficiency. Furthermore, we apply the optimal production transport model to the coal production and transport problem. Numerical simulation demonstrates that our proposed model can save the production and transport cost by 13.17%.

A molecular beam epitaxial (MBE) model with slope selection consisting of a fourth order Ginzburg-Landau double well potential, is derived from the variation of the free energy. One challenge in constructing the local structure preserving algorithms (SPAs) for the MBE model with slope selection is how to properly discretize the equation in space and time simultaneously in order to preserve the local structure at the discrete level. To resolve this issue, we employ the local energy dissipation property and the energy quadratization techniques. One novelty is that all nonlinear terms are treated semi-explicitly. The other novelty is that we introduce proper intermediate variables to make the space operators act on one single term, which is one crucial step in constructing local SPAs. We then develop two local energy dissipation preserving schemes and show rigorously the local energy dissipation property of the two schemes. Under suitable boundary conditions, such as periodic boundary conditions, the algorithms can preserve not only mass but also global energy dissipation property. Numerical experiments confirm the second-order accuracy and show the excellent performance of the schemes proposed.

Subdiffusion equations with distributed-order fractional derivatives describe important physical phenomena. In this paper, we consider an inverse space-dependent source term problem governed by a distributed order time-fractional diffusion equation using final time data. Based on the series expression of the solution, the inverse source problem can be transformed into a first kind of Fredholm integral equation. The existence, uniqueness and a conditional stability of the considered inverse problem are established. Building upon this foundation, a generalized quasi-boundary value regularization method is proposed to solve the inverse source problem, and then we prove the well-posedness of the regularized problem. Further, we provide the convergence rates of the regularized solution by employing an a priori and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to validate the effectiveness of the proposed method.

The main objective of this study is to present a fast and efficient numerical scheme for solving nonlinear integral differential equations with weak singular kernels in three-dimensional domain. First, the temporal derivative and integral term are approximated by the Crank-Nicolson (CN) method and the second-order fractional quadrature rule. After that the spatial discretization is carried out by combining the finite difference method and the alternating direction implicit (ADI) method, and the nonlinear term are approximated using the Taylor expansion. The stability and convergence of the proposed scheme are analyzed, followed by the verification of the theoretical results through numerical experiments.