arXiv subjects

Guangtao Zhang

Publications and source records attributed to Guangtao Zhang.

5 recordsLinked to original sources

arXiv2026

Inverse Problems for Partial Differential Equations with Jump Discontinuities in Coefficients via Two-Stage Physics-Informed Deep Learning and Statistical Mixture Models

This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main network approximates the PDE solution and an auxiliary coefficient sub network provides a relaxed continuous surrogate of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated into the physics residual to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which estimate the number of coefficient regimes and provide heuristic search intervals for coefficient values and candidate transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is represented explicitly as a hard piecewise-constant function over the spatiotemporal domain. Numerical experiments on different PDE types with jump-discontinuous coefficients demonstrate that the proposed framework achieves accurate parameter estimation with acceptable computational costs compared to existing methods. This work provides an effective integrated workflow for inverse problems governed by PDEs with discontinuous parameter structures, particularly in nonstationary and heterogeneous systems.

stat.ML↗

arXiv2026

Physics-Consistent Diffusion for Efficient Fluid Super-Resolution via Multiscale Residual Correction

Existing image SR and generic diffusion models transfer poorly to fluid SR: they are sampling-intensive, ignore physical constraints, and often yield spectral mismatch and spurious divergence. We address fluid super-resolution (SR) with \textbf{ReMD} (\underline{Re}sidual-\underline{M}ultigrid \underline{D}iffusion), a physics-consistent diffusion framework. At each reverse step, ReMD performs a \emph{multigrid residual correction}: the update direction is obtained by coupling data consistency with lightweight physics cues and then correcting the residual across scales; the multiscale hierarchy is instantiated with a \emph{multi-wavelet} basis to capture both large structures and fine vortical details. This coarse-to-fine design accelerates convergence and preserves fine structures while remaining equation-free. Across atmospheric and oceanic benchmarks, ReMD improves accuracy and spectral fidelity, reduces divergence, and reaches comparable quality with markedly fewer sampling steps than diffusion baselines. Our results show that enforcing physics consistency \emph{inside} the diffusion process via multigrid residual correction and multi-wavelet multiscale modeling is an effective route to efficient fluid SR. Our code are available on https://github.com/lizhihao2022/ReMD.

cs.CV↗

arXiv2026

A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations

Physics-Informed Neural Networks (PINNs) encounter accuracy limitations when solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). To overcome this, we employ a novel loss function, Residuals-weighted Region Activation Evaluation (Residuals-RAE), featuring a { pre-training weight update scheme}. { Unlike conventional self-adaptive PINNs where weights evolve simultaneously with network parameters, Residuals-RAE-PINNs computes weights from current residuals before each training step and holds them constant during gradient updates. We establish weight convergence under standard neural network optimization assumptions, which justifies analyzing the converged network with constant weights.} Based on this theoretical framework, we derive the error estimation for PINNs with Residuals-RAE when solving AC and CH equations. {The analysis is aligned with Monte-Carlo sampling for the discretization of integrals, consistent with the numerical experiments.} Numerical experiments on one- and two-dimensional AC and CH systems confirm our theoretical results. Additionally, our analysis reveals that feedforward neural networks with two hidden layers and the tanh activation function bound the approximation errors of the solution, its temporal derivative, and the nonlinear term, constrained by the training loss and the number of collocation points.

math.NA↗

arXiv2025

Feel the Difference? A Comparative Analysis of Emotional Arcs in Real and LLM-Generated CBT Sessions

Synthetic therapy dialogues generated by large language models (LLMs) are increasingly used in mental health NLP to simulate counseling scenarios, train models, and supplement limited real-world data. However, it remains unclear whether these synthetic conversations capture the nuanced emotional dynamics of real therapy. In this work, we introduce RealCBT, a dataset of authentic cognitive behavioral therapy (CBT) dialogues, and conduct the first comparative analysis of emotional arcs between real and LLM-generated CBT sessions. We adapt the Utterance Emotion Dynamics framework to analyze fine-grained affective trajectories across valence, arousal, and dominance dimensions. Our analysis spans both full dialogues and individual speaker roles (counselor and client), using real sessions from the RealCBT dataset and synthetic dialogues from the CACTUS dataset. We find that while synthetic dialogues are fluent and structurally coherent, they diverge from real conversations in key emotional properties: real sessions exhibit greater emotional variability, more emotion-laden language, and more authentic patterns of reactivity and regulation. Moreover, emotional arc similarity remains low across all pairings, with especially weak alignment between real and synthetic speakers. These findings underscore the limitations of current LLM-generated therapy data and highlight the importance of emotional fidelity in mental health applications. To support future research, our dataset RealCBT is released at https://gitlab.com/xiaoyi.wang/realcbt-dataset.

cs.CL↗

arXiv2023

Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients

To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.

cs.LG↗