arXiv subjects

Lianzi Jiang

Publications and source records attributed to Lianzi Jiang.

14 recordsLinked to original sources

arXiv2026

Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem

This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds.

math.PR↗

arXiv2025

A Physics Prior-Guided Dual-Stream Attention Network for Motion Prediction of Elastic Bragg Breakwaters

Accurate motion response prediction for elastic Bragg breakwaters is critical for their structural safety and operational integrity in marine environments. However, conventional deep learning models often exhibit limited generalization capabilities when presented with unseen sea states. These deficiencies stem from the neglect of natural decay observed in marine systems and inadequate modeling of wave-structure interaction (WSI). To overcome these challenges, this study proposes a novel Physics Prior-Guided Dual-Stream Attention Network (PhysAttnNet). First, the decay bidirectional self-attention (DBSA) module incorporates a learnable temporal decay to assign higher weights to recent states, aiming to emulate the natural decay phenomenon. Meanwhile, the phase differences guided bidirectional cross-attention (PDG-BCA) module explicitly captures the bidirectional interaction and phase relationship between waves and the structure using a cosine-based bias within a bidirectional cross-computation paradigm. These streams are synergistically integrated through a global context fusion (GCF) module. Finally, PhysAttnNet is trained with a hybrid time-frequency loss that jointly minimizes time-domain prediction errors and frequency-domain spectral discrepancies. Comprehensive experiments on wave flume datasets demonstrate that PhysAttnNet significantly outperforms mainstream models. Furthermore,cross-scenario generalization tests validate the model's robustness and adaptability to unseen environments, highlighting its potential as a framework to develop predictive models for complex systems in ocean engineering.

cs.LG↗

arXiv2025

A Causality- and Frequency-Aware Deep Learning Framework for Wave Elevation Prediction Behind Floating Breakwaters

Predicting the elevations of nonlinear wave fields behind floating breakwaters (FBs) is crucial for optimizing coastal engineering structures, enhancing safety, and improving design efficiency. Existing deep learning approaches exhibit limited generalization capability under unseen operating conditions. To address this challenge, this study proposes the Exogenous-to-Endogenous Frequency-Aware Network (E2E-FANet), a novel end-to-end neural network designed to model relationships between waves and structures. First, the Dual-Basis Frequency Mapping (DBFM) module leverages orthogonal cosine and sine bases to generate an adaptive time-frequency representation, enabling the model to effectively disentangle the evolving spectral components of wave signals. Second, the Exogenous-to-Endogenous Cross-Attention (E2ECA) module employs cross attention to explicitly model the unidirectional causal influence of floating breakwater motion on wave elevations. Additionally, a Temporal-wise Attention (TA) mechanism is incorporated that adaptively captures complex dependencies in endogenous variables. Extensive experiments, including generalization tests across diverse wave conditions and adaptability tests under varying relative water density (RW) conditions, demonstrate that E2E-FANet achieves superior predictive accuracy and robust generalization compared to mainstream models. This work emphasizes the importance of integrating causality and frequency-aware modeling in deep learning architectures for modeling nonlinear dynamics systems.

cs.LG↗

arXiv2025

A Novel Framework for Significant Wave Height Prediction based on Adaptive Feature Extraction Time-Frequency Network

Precise forecasting of significant wave height (Hs) is essential for the development and utilization of wave energy. The challenges in predicting Hs arise from its non-linear and non-stationary characteristics. The combination of decomposition preprocessing and machine learning models have demonstrated significant effectiveness in Hs prediction by extracting data features. However, decomposing the unknown data in the test set can lead to data leakage issues. To simultaneously achieve data feature extraction and prevent data leakage, a novel Adaptive Feature Extraction Time-Frequency Network (AFE-TFNet) is proposed to improve prediction accuracy and stability. It is encoder-decoder rolling framework. The encoder consists of two stages: feature extraction and feature fusion. In the feature extraction stage, global and local frequency domain features are extracted by combining Wavelet Transform (WT) and Fourier Transform (FT), and multi-scale frequency analysis is performed using Inception blocks. In the feature fusion stage, time-domain and frequency-domain features are integrated through dominant harmonic sequence energy weighting (DHSEW). The decoder employed an advanced long short-term memory (LSTM) model. Hourly measured wind speed (Ws), dominant wave period (DPD), average wave period (APD) and Hs from three stations are used as the dataset, and the four metrics are employed to evaluate the forecasting performance. Results show that AFE-TFNet significantly outperforms benchmark methods in terms of prediction accuracy. Feature extraction can significantly improve the prediction accuracy. DHSEW has substantially increased the accuracy of medium-term to long-term forecasting. The prediction accuracy of AFE-TFNet does not demonstrate significant variability with changes of rolling time window size. Overall, AFE-TFNet shows strong potential for handling complex signal forecasting.

cs.LG↗

arXiv2024

Discrete-time approximation for backward stochastic differential equations driven by $G$-Brownian motion

In this paper, we study the discrete-time approximation schemes for a class of backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) which corresponds to the hedging pricing of European contingent claims. By introducing an auxiliary extended $\widetilde{G}$-expectation space, we propose a class of $θ$-schemes to discrete $G$-BSDEs in this space. With the help of nonlinear stochastic analysis techniques and numerical analysis tools, we prove that our schemes admit half-order convergence for approximating $G$-BSDE in the general case. In some special cases, our schemes can achieve a first-order convergence rate. Finally, we give an implementable numerical scheme for $G$-BSDEs based on Peng's central limit theorem and illustrate our convergence results with numerical examples.

math.NA↗

arXiv2024

On the rate of convergence for an $α$-stable central limit theorem under sublinear expectation

In this paper, we propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations (PIDEs) which characterize the nonlinear $α$-stable Lévy processes under sublinear expectation space with $α\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry-Esseen bound and convergence rate for the $α$-stable central limit theorem under sublinear expectation.

math.PR↗

arXiv2023

Convergence rates for Chernoff-type approximations of convex monotone semigroups

We provide explicit convergence rates for Chernoff-type approximations of convex monotone semigroups which have the form $S(t)f=\lim_{n\to\infty}I(\frac{t}{n})^n f$ for bounded continuous functions $f$. Under suitable conditions on the one-step operators $I(t)$ regarding the time regularity and consistency of the approximation scheme, we obtain $\|S(t)f-I(\frac{t}{n})^n f\|_\infty\leq cn^{-γ}$ for bounded Lipschitz continuous functions $f$, where $c\geq 0$ and $γ>0$ are determined explicitly. Moreover, the mapping $t\mapsto S(t)f$ is Hölder continuous. These results are closely related to monotone approximation schemes for viscosity solutions but are obtained independently by following a recently developed semigroup approach to Hamilton-Jacobi-Bellman equations which uniquely characterizes semigroups via their $Γ$-generators. The different approach allows to consider convex rather than sublinear equations and the results can be extended to unbounded functions by modifying the norm with a suitable weight function. Furthermore, up to possibly different consistency errors for the operators $I(t)$, the upper and lower bound for the error between the semigroup and the iterated operators are symmetric. The abstract results are applied to Nisio semigroups and limit theorems for convex expectations.

math.PR↗

arXiv2023

Error estimates for the robust $α$-stable central limit theorem under sublinear expectation by discrete approximation method

In this work, we develop a numerical method to study the error estimates of the $α$-stable central limit theorem under sublinear expectation with $α\in(0,2)$, whose limit distribution can be characterized by a fully nonlinear integro-differential equation (PIDE). Based on the sequence of independent random variables, we propose a discrete approximation scheme for the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis techniques and numerical analysis tools, we establish the error bounds for the discrete approximation scheme, which in turn provides a general error bound for the robust $α$-stable central limit theorem, including the integrable case $α\in(1,2)$ as well as the non-integrable case $α\in(0,1]$. Finally, we provide some concrete examples to illustrate our main results and derive the precise convergence rates.

math.NA↗

arXiv2023

A robust $α$-stable central limit theorem under sublinear expectation without integrability condition

This article relaxes the integrability condition imposed in the literature for the robust $α$-stable central limit theorem under sublinear expectation. Specifically, for $α\in(0,1]$, we prove that the normalized sums of i.i.d. non-integrable random variables $\big \{n^{-\frac{1}α}\sum_{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty}$ converge in distribution to $\tildeζ_{1}$, where $(\tildeζ_{t})_{t\in \lbrack0,1]}$ is a multidimensional nonlinear symmetric $α$-stable process with a jump uncertainty set $\mathcal{L}$. The limiting $α$-stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l}\displaystyle \partial_{t}u(t,x)-\sup \limits_{F_μ\in \mathcal{L}}\left \{ \int_{\mathbb{R}^{d}}δ_λ^αu(t,x)F_μ(dλ)\right \} =0,\\ \displaystyle u(0,x)=ϕ(x),\ \ \ \forall(t,x)\in \lbrack0,1]\times \mathbb{R}^{d}, \end{array} \right. \] where \[ δ_λ^α u(t,x):= \left \{ \begin{array} [c]{l} u(t,x+λ)-u(t,x)-\langle D_{x}u(t,x),λ\mathbb{1}_{\{|λ|\leq 1\}}\rangle,\ α=1,\\ u(t,x+λ)-u(t,x),\ α\in(0,1). \end{array} \right. \] The main tools are a weak convergence approach to obtain the limiting process, a Lévy-Khintchine representation of the nonlinear $α$-stable process and a truncation technique to estimate the corresponding $α$-stable Lévy measures. As a byproduct, the article also provides a probabilistic approach to prove the existence of the above fully nonlinear PIDE.

math.PR↗

arXiv2022

A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation

This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $α\in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[α]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde ξ_{t},\tildeη_{t},\tildeζ_{t})$, $t\in [0,1]$, is a multidimensional nonlinear Lévy process with an uncertainty set $Θ$ as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l} \displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_μ,q,Q)\in Θ}\left \{ \int_{\mathbb{R}^{d}}δ_λu(t,x,y,z)F_μ(dλ)\right. \\ \displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ \displaystyle u(0,x,y,z)=ϕ(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right. \] with $δ_λu(t,x,y,z):=u(t,x,y,z+λ)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),λ\rangle$. To construct the limit process $(\tilde{L}_{t})_{t\in \lbrack0,1]}$, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize $(\tilde{L}_{t})_{t\in [0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

math.PR↗

arXiv2022

An Efficient Numerical Method for Forward-Backward Stochastic Differential Equations Driven by $G$-Brownian motion

In this paper, we study the numerical method for solving forward-backward stochastic differential equations driven by $G$-Brownian motion ($G$-FBSDEs) which correspond to fully nonlinear partial differential equations (PDEs). First, we give an approximate conditional $G$-expectation and obtain feasible methods to calculate the distribution of $G$-Brownian motion. On this basis, some efficient numerical schemes for $G$-FBSDEs are then proposed. We rigorously analyze errors of the proposed schemes and prove the convergence results. Finally, several numerical experiments are given to demonstrate the accuracy of our method.

math.NA↗

arXiv2021

Discrete-time approximation for stochastic optimal control problems under the $G$-expectation framework

In this paper, we propose a class of discrete-time approximation schemes for stochastic optimal control problems under the $G$-expectation framework. The proposed schemes are constructed recursively based on piecewise constant policy. We prove the convergence of the discrete schemes and determine the convergence rates. Several numerical examples are presented to illustrate the effectiveness of the obtained results.

math.OC↗

arXiv2020

An Effective Discrete Recursive Method for Stochastic Optimal Control Problems

In this paper, we study the numerical method for stochastic optimal control problems (SOCPs). By reducing the optimal control problem to the discrete case, we derive a discrete stochastic maximum principle (SMP). With the help of this SMP, we propose an effective discrete recursive method for SOCPs with feedback control. We rigorously analyze errors of the proposed method and prove that the cost obtained by our method is of first-order convergence. Numerical experiments are carried out to support our theoretical results.

math.NA↗

arXiv2019

Explicit $θ$-Schemes for Solving Anticipated Backward Stochastic Differential Equations

In this paper, a class of stable explicit $θ$-schemes are proposed for solving anticipated backward stochastic differential equations (anticipated BSDEs) which generator not only contains the present values of the solutions but also the future. We subtly transform the delay process of the generator into the current measurable process, resulting in high-order convergence rate. We also analyze the stability of our numerical schemes and strictly prove the error estimates. Various numerical tests powerful demonstrate high accuracy of the proposed numerical schemes.

math.NA↗