arXiv subjects

Qiaolin He

Publications and source records attributed to Qiaolin He.

At least 19 recordsLinked to original sources

arXiv2026

DeSyR: A Decoupled Symbolic Recovery Framework with PINN-Guided Structure Search and Physics-Informed Coefficient Refinement

Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an $O(\beta^{-1})$ teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative $L_2$ errors are $2.31\times10^{-14}$ or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.

cs.LG↗

arXiv2026

Stability of admissible solutions for coexisting phase transitions for one-dimensional compressible van der Waals fluids

In this paper, we investigate the dynamic stability of certain steady-state solutions to the periodic boundary value problem for compressible isentropic Navier-Stokes system under the van der Waals equation of state in one space dimension. These steady-state solutions correspond to the admissible solutions describing two-phase coexisting phase transitions, where the integral average of the specific volume belongs to the Maxwell region. We first construct a semi-discrete staggered grid difference scheme to prove the local existence of solutions to the periodic problem, without imposing the standard stability hypothesis $p_v<0$. Then, by virtue of rigorous piecewise a priori estimates, we demonstrate that the periodic boundary value problem for van der Waals fluids possesses a global solution existing for all time, and this solution converges uniformly to the admissible steady state as time tends to infinity. This result firmly establishes the nonlinear stability of the admissible phase-transition solutions under general small initial disturbances.

math.AP↗

arXiv2026

A Novel Fourier Feature Network for Solving Partial Differential Equations

Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both. Similar to Extreme Learning Machines (ELMs), FENs employ a single-hidden-layer architecture to generate a set of basis functions. The target function is then approximated as a linear combination of these basis functions, with the coefficients determined using the least squares method. However, unlike ELMs, which often rely on affine transformations to improve representational power, FENs can achieve high-precision solutions without requiring such transformations on the input variables. To evaluate the representational capacity of these networks, we search for an optimal scaling factor within a predefined range for the randomly initialized and fixed weights and biases. By adjusting this scaling factor, we ensure a fair comparison between FENs and ELMs using various activation functions, such as $\text{sigmoid}$, $\tanh$, and $\text{swish}$. Our numerical experiments demonstrate that FENs consistently achieve higher accuracy than ELMs.

cs.LG↗

arXiv2026

Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs

Solving high-frequency partial differential equations (PDEs) with neural networks is notoriously difficult due to the spectral bias of conventional architectures. We propose the Multi-Scale Separable Fourier Neural Network (MS-SFNN), a framework designed to overcome this limitation by explicitly encoding multi-scale Fourier features within a separable representation. The network factorizes the solution into $d$ single-coordinate subnetworks with fixed, randomly initialized weights; these subnetworks are combined via element-wise products to form a rich set of basis functions. This separable construction scales linearly with the problem dimension, thus inherently alleviating the curse of dimensionality. Crucially, each subnetwork is equipped with trainable scaling factors coupled with cosine activations, providing an adaptive mechanism for multi-scale frequency selection that endows the model with strong spectral approximation capability. The PDE solution is expressed as a linear combination of these learned basis functions, and the combination coefficients are determined by solving a large-scale least-squares system. To resolve the memory bottleneck in high-frequency or three-dimensional settings, we replace automatic differentiation (AD) with analytical derivatives of the basis functions and use a memory-efficient batched QR decomposition for solving the least-squares systems efficiently. Extensive numerical experiments demonstrate that MS-SFNN achieves superior accuracy and substantially outperforms state-of-the-art methods, including Physics-Informed Neural Network (PINN) and the Separated-Variable Spectral Neural Network (SV-SNN).

cs.LG↗

arXiv2026

A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

cs.LG↗

arXiv2026

A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations

In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2026

A Regularized Framework and Admissible Solutions for Liquid-Vapor Phase Transitions in Steady Compressible Flows

We investigate the well-posedness of the periodic boundary value problem for the steady compressible isentropic Navier-Stokes system under the van der Waals equation of state. The main difficulty arises from the non-monotonicity of the pressure, which induces liquid-vapor phase transitions and consequently leads to both physical instabilities and mathematical non-uniqueness of solutions. It is shown that the occurrence of a phase transition is determined by whether the integral average of the specific volume lies inside the gas-liquid coexistence region defined by the Maxwell construction. By introducing an artificial viscosity, we construct an approximate system. When the integral average of the specific volume falls within the Maxwell region, the approximate solution converges, as the artificial viscosity tends to zero, to the equilibrium states given by Maxwell's construction, with the diffuse interface sharpening into a discontinuity. Conversely, if the integral average of the specific volume lies outside this region, the limiting solution remains outside as well, meaning that no phase transition occurs. These results demonstrate that the non-monotonicity of the pressure, combined with the condition that the integral average of the specific volume belongs to the Maxwell region, can act as a nucleation mechanism for phase transitions in the isentropic gas-liquid problem. Furthermore, the proposed approximation not only offers a regularized framework for describing phase transitions but also provides, from a rigorous mathematical viewpoint, a definition of admissible solutions related to phase transitions. The detailed proof relies on the artificial viscosity method, the calculus of variations, the anti-derivative technique, phase-plane analysis, and the level-set method.

math.AP↗

arXiv2026

CyIN: Cyclic Informative Latent Space for Bridging Complete and Incomplete Multimodal Learning

Multimodal machine learning, mimicking the human brain's ability to integrate various modalities has seen rapid growth. Most previous multimodal models are trained on perfectly paired multimodal input to reach optimal performance. In real-world deployments, however, the presence of modality is highly variable and unpredictable, causing the pre-trained models in suffering significant performance drops and fail to remain robust with dynamic missing modalities circumstances. In this paper, we present a novel Cyclic INformative Learning framework (CyIN) to bridge the gap between complete and incomplete multimodal learning. Specifically, we firstly build an informative latent space by adopting token- and label-level Information Bottleneck (IB) cyclically among various modalities. Capturing task-related features with variational approximation, the informative bottleneck latents are purified for more efficient cross-modal interaction and multimodal fusion. Moreover, to supplement the missing information caused by incomplete multimodal input, we propose cross-modal cyclic translation by reconstruct the missing modalities with the remained ones through forward and reverse propagation process. With the help of the extracted and reconstructed informative latents, CyIN succeeds in jointly optimizing complete and incomplete multimodal learning in one unified model. Extensive experiments on 4 multimodal datasets demonstrate the superior performance of our method in both complete and diverse incomplete scenarios.

cs.LG↗

arXiv2026

MissMAC-Bench: Building Solid Benchmark for Missing Modality Issue in Robust Multimodal Affective Computing

As a knowledge discovery task over heterogeneous data sources, current Multimodal Affective Computing (MAC) heavily rely on the completeness of multiple modalities to accurately understand human's affective state. However, in real-world scenarios, the availability of modality data is often dynamic and uncertain, leading to substantial performance fluctuations due to the distribution shifts and semantic deficiencies of the incomplete multimodal inputs. Known as the missing modality issue, this challenge poses a critical barrier to the robustness and practical deployment of MAC models. To systematically quantify this issue, we introduce MissMAC-Bench, a comprehensive benchmark designed to establish fair and unified evaluation standards from the perspective of cross-modal synergy. Two guiding principles are proposed, including no missing prior during training, and one single model capable of handling both complete and incomplete modality scenarios, thereby ensuring better generalization. Moreover, to bridge the gap between academic research and real-world applications, our benchmark integrates evaluation protocols with both fixed and random missing patterns at the dataset and instance levels. Extensive experiments conducted on 3 widely-used language models across 4 datasets validate the effectiveness of diverse MAC approaches in tackling the missing modality issue. Our benchmark provides a solid foundation for advancing robust multimodal affective computing and promotes the development of multimedia data mining.

cs.AI↗

arXiv2026

Architecture-Optimization Co-Design for Physics-Informed Neural Networks Via Attentive Representations and Conflict-Resolved Gradients

Physics-Informed Neural Networks (PINNs) provide a learning-based framework for solving partial differential equations (PDEs) by embedding governing physical laws into neural network training. In practice, however, their performance is often hindered by limited representational capacity and optimization difficulties caused by competing physical constraints and conflicting gradients. In this work, we study PINN training from a unified architecture-optimization perspective. We first propose a layer-wise dynamic attention mechanism to enhance representational flexibility, resulting in the Layer-wise Dynamic Attention PINN (LDA-PINN). We then reformulate PINN training as a multi-task learning problem and introduce a conflict-resolved gradient update strategy to alleviate gradient interference, leading to the Gradient-Conflict-Resolved PINN (GC-PINN). By integrating these two components, we develop the Architecture-Conflict-Resolved PINN (ACR-PINN), which combines attentive representations with conflict-aware optimization while preserving the standard PINN loss formulation. Extensive experiments on benchmark PDEs, including the Burgers, Helmholtz, Klein-Gordon, and lid-driven cavity flow problems, demonstrate that ACR-PINN achieves faster convergence and significantly lower relative $L_2$ and $L_\infty$ errors than standard PINNs. These results highlight the effectiveness of architecture-optimization co-design for improving the robustness and accuracy of PINN-based solvers.

cs.LG↗

arXiv2025

Thermodynamically consistent modeling and simulation of two-fluid magnetohydrodynamic equations

Based on a rigorous thermodynamic framework, this work develops a two-fluid magnetohydrodynamic model grounded in the Helmholtz free energy formalism. The model maintains full thermodynamic consistency by simultaneously satisfying energy conservation and entropy production laws in two-fluid systems. By analyzing the convex-concave structure of the Helmholtz free energy density, we systematically derive key thermodynamic variables-chemical potential, entropy density, and internal energy-in a self-consistent manner. Building on this foundation, we construct a temporally discrete numerical scheme that inherits the thermodynamic consistency of the continuous model. The scheme is proven to adhere rigorously to both the first and second laws of thermodynamics. For the implemented two-dimensional degenerate system, we establish comprehensive a priori error estimates in space and time. Numerical simulations validate the model's effectiveness in capturing essential plasma phenomena, demonstrating its applicability to complex physical scenarios.

physics.plasm-ph↗

arXiv2025

Multi-source Multimodal Progressive Domain Adaption for Audio-Visual Deception Detection

This paper presents the winning approach for the 1st MultiModal Deception Detection (MMDD) Challenge at the 1st Workshop on Subtle Visual Computing (SVC). Aiming at the domain shift issue across source and target domains, we propose a Multi-source Multimodal Progressive Domain Adaptation (MMPDA) framework that transfers the audio-visual knowledge from diverse source domains to the target domain. By gradually aligning source and the target domain at both feature and decision levels, our method bridges domain shifts across diverse multimodal datasets. Extensive experiments demonstrate the effectiveness of our approach securing Top-2 place. Our approach reaches 60.43% on accuracy and 56.99\% on F1-score on competition stage 2, surpassing the 1st place team by 5.59% on F1-score and the 3rd place teams by 6.75% on accuracy. Our code is available at https://github.com/RH-Lin/MMPDA.

cs.CV↗

arXiv2025

E3RG: Building Explicit Emotion-driven Empathetic Response Generation System with Multimodal Large Language Model

Multimodal Empathetic Response Generation (MERG) is crucial for building emotionally intelligent human-computer interactions. Although large language models (LLMs) have improved text-based ERG, challenges remain in handling multimodal emotional content and maintaining identity consistency. Thus, we propose E3RG, an Explicit Emotion-driven Empathetic Response Generation System based on multimodal LLMs which decomposes MERG task into three parts: multimodal empathy understanding, empathy memory retrieval, and multimodal response generation. By integrating advanced expressive speech and video generative models, E3RG delivers natural, emotionally rich, and identity-consistent responses without extra training. Experiments validate the superiority of our system on both zero-shot and few-shot settings, securing Top-1 position in the Avatar-based Multimodal Empathy Challenge on ACM MM 25. Our code is available at https://github.com/RH-Lin/E3RG.

cs.AI↗

arXiv2025

Thermodynamically consistent modelling and simulation of the moving contact line problem in non-isothermal compressible two-phase flows

According to the dynamic van der Waals theory, we propose a thermodynamically consistent model for non-isothermal compressible two-phase flows with contact line motion. In this model, fluid temperature is treated as a primary variable, characterized by the proposed temperature equation instead of being obtained from intermediate variables such as total energy density, internal energy density and entropy density. The hydrodynamic boundary conditions, which represent a generalization of the generalized Navier slip boundary condition in non-isothermal flows, are imposed on the proposed model. We then develop the dimensionless form of the model and prove that it rigorously satisfies the first and second laws of thermodynamics. Two numerical schemes based on the dimensionless system are constructed: one is fully coupled and thermodynamically consistent, namely strictly satisfying the temporally discrete first and second laws of thermodynamics; the other, designed by extending the multiple scalar auxiliary variable approach to entropy production, is decoupled, linear, and unconditionally entropy-stable. Several numerical results are presented to validate the effectiveness and stability of the proposed method.

physics.flu-dyn↗

arXiv2025

Adaptive feature capture method for solving partial differential equations with near singular solutions

Partial differential equations (PDEs) with near singular solutions pose significant challenges for traditional numerical methods, particularly in complex geometries where mesh generation and adaptive refinement become computationally expensive. Although deep-learning-based approaches, such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM), offer mesh-free alternatives, they often lack adaptive resolution in critical regions, limiting their accuracy for solutions with steep gradients or singularities. In this work, we propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high-gradient regions to enhance local expressive power. Inspired by adaptive moving mesh techniques, AFCM uses the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead. The AFCM extends the capabilities of RFM to handle PDEs with near-singular solutions while preserving its mesh-free efficiency. Numerical experiments demonstrate the method's effectiveness in accurately resolving near-singular problems with a performance that is better than that of the traditional finite element method in terms of accuracy and efficiency. AFCM offers a robust and scalable approach to solving challenging PDEs in scientific and engineering applications.

math.NA↗

arXiv2025

A novel number-theoretic sampling method for neural network solutions of partial differential equations

Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics-Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low-regularity Poisson equations, two-dimensional inverse Helmholtz problems, and high-dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities.

math.NA↗

arXiv2025

The Cauchy Problem for Non-Isentropic compressible Navier-Stokes/Allen-Cahn system with Degenerate Heat-Conductivity

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity $κ(θ)=\tildeκθ^β$ in 1-d is discussed in this paper. This system is widely used to describe the motion of immiscible two-phase flow in numerical simulation. The wellposedness for strong solution of this problem is established with the $H^1$ initial data for density, temperature, velocity, and the $H^2$ initial data for phase field. The result shows that no discontinuity of the phase field, vacuum, shock wave, mass or heat concentration will be developed at any finite time in the whole space. From the hydrodynamic point of view, this means that no matter how complex the interaction between the hydrodynamic and phase-field effects, phase separation will not occur, but the phase transition is possible.

math.AP↗

arXiv2024

Runge-Kutta Random Feature Method for Solving Multiphase Flow Problems of Cells

Cell collective migration plays a crucial role in a variety of physiological processes. In this work, we propose the Runge-Kutta random feature method to solve the nonlinear and strongly coupled multiphase flow problems of cells, in which the random feature method in space and the explicit Runge-Kutta method in time are utilized. Experiments indicate that this algorithm can effectively deal with time-dependent partial differential equations with strong nonlinearity, and achieve high accuracy both in space and time. Moreover, in order to improve computational efficiency and save computational resources, we choose to implement parallelization and non-automatic differentiation strategies in our simulations. We also provide error estimates for the Runge-Kutta random feature method, and a series of numerical experiments are shown to validate our method.

math.NA↗