arXiv subjects

Yuhuan Yuan

Publications and source records attributed to Yuhuan Yuan.

At least 19 recordsLinked to original sources

arXiv2026

Sample-Efficient Post-Training for LEGO Spatial-Physics Reasoning

LLM-based LEGO assembly requires both semantic grounding and physical feasibility. In this paper, we identify a data-induced failure mode, physhack, in which generated assemblies satisfy physical-validity constraints while remaining geometrically misaligned, semantically inconsistent, or poorly calibrated. To address this challenge, we propose a model-based data selection approach that uses only 5% of the original training data while improving semantic and structural alignment across multiple independent evaluators. We further introduce PVPO, a reinforcement learning method that couples physical-validity and voxel-space geometric rewards to strengthen the LEGO synthesis capabilities of LLMs. PVPO-trained DeepSeek-Llama-8B outperforms the frontier VLM Kimi-K3 across semantic, structural, and physical evaluation dimensions, even when Kimi-K3 is provided with a ground-truth image as an additional reference.

cs.LG↗

arXiv2026

Quantum algorithms for Young measures: applications to nonlinear partial differential equations

Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction.

quant-ph↗

arXiv2026

Temperature-driven turbulence in compressible fluid flows

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

math.NA↗

arXiv2026

Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.

math.NA↗

arXiv2025

Convergence analysis for a finite volume evolution Galerkin method for multidimensional hyperbolic systems

We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of gas dynamics. We show the stability and the consistency of the numerical approximations. By means of the generalized Lax equivalence principle we prove the convergence of numerical solutions to the strong solution on the lifespan.

math.NA↗

arXiv2025

A note on relations between convexity and concavity of thermodynamic functions

The paper is concerned with proving the equivalence of convexity or concavity properties of thermodynamic functions, such as energy and entropy, depending on different sets of variables. These variables are the basic thermodynamic state variables, specific state variables or the densities of state variables that are used in continuum mechanics. We prove results for transformations of variables and functions in conjunction with convexity properties. We are concerned with convexity, strict convexity, positive definite Hessian matrices and the analogous forms of concavity. The main results are equivalence relations for these properties between functions. These equivalences are independent of the equations of state since they only use general properties of them. The results can be used for instance to easily prove that the entropy density function for the Euler equations in conservative variables in three space dimensions is strictly concave or even has a negative definite Hessian matrix. Further, we show how various equations of state imply these properties and how these properties are relevant to mathematical analysis.

math.AP↗

arXiv2025

Generating Symbolic World Models via Test-time Scaling of Large Language Models

Solving complex planning problems requires Large Language Models (LLMs) to explicitly model the state transition to avoid rule violations, comply with constraints, and ensure optimality-a task hindered by the inherent ambiguity of natural language. To overcome such ambiguity, Planning Domain Definition Language (PDDL) is leveraged as a planning abstraction that enables precise and formal state descriptions. With PDDL, we can generate a symbolic world model where classic searching algorithms, such as A*, can be seamlessly applied to find optimal plans. However, directly generating PDDL domains with current LLMs remains an open challenge due to the lack of PDDL training data. To address this challenge, we propose to scale up the test-time computation of LLMs to enhance their PDDL reasoning capabilities, thereby enabling the generation of high-quality PDDL domains. Specifically, we introduce a simple yet effective algorithm, which first employs a Best-of-N sampling approach to improve the quality of the initial solution and then refines the solution in a fine-grained manner with verbalized machine learning. Our method outperforms o1-mini by a considerable margin in the generation of PDDL domains, achieving over 50\% success rate on two tasks (i.e., generating PDDL domains from natural language description or PDDL problems). This is done without requiring additional training. By taking advantage of PDDL as state abstraction, our method is able to outperform current state-of-the-art methods on almost all competition-level planning tasks.

cs.AI↗

arXiv2024

Error analysis of the Monte Carlo method for compressible magnetohydrodynamics

We study random compressible viscous magnetohydrodynamic flows. Combining the Monte Carlo method with a deterministic finite volume method we solve the random system numerically. Quantitative error estimates including statistical and deterministic errors are analyzed up to a stopping time of the exact solution. On the life-span of an exact strong solution we prove the convergence of the numerical solutions. Numerical experiments illustrate rich dynamics of random viscous compressible magnetohydrodynamics.

math.NA↗

arXiv2024

Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

math.NA↗

arXiv2024

What is a limit of structure-preserving numerical methods for compressible flows?

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.

math.NA↗

arXiv2024

Convergence of numerical methods for the Navier-Stokes-Fourier system driven by uncertain initial/boundary data

We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem.

math.NA↗

arXiv2024

Convergence of a generalized Riemann problem scheme for the Burgers equation

In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.

math.NA↗

arXiv2023

Penalty method for the Navier-Stokes-Fourier system with Dirichlet boundary conditions: convergence and error estimates

We study the convergence and error estimates of a finite volume method for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions. Physical fluid domain is typically smooth and needs to be approximated by a polygonal computational domain. This leads to domain-related discretization errors, the so-called variational crimes. To treat them efficiently we embed the fluid domain into a large enough cubed domain, and propose a finite volume scheme for the corresponding domain-penalized problem. Under the assumption that the numerical density and temperature are uniformly bounded, we derive the ballistic energy inequality, yielding a priori estimates and the consistency of the penalization finite volume approximations. Further, we show that the numerical solutions converge weakly to a generalized, the so-called dissipative measure-valued, solution of the corresponding Dirichlet problem. If a strong solution exists, we prove that our numerical approximations converge strongly with the rate 1/4. Additionally, assuming uniform boundedness of the approximate velocities, we obtain global existence of the strong solution. In this case we prove that the numerical solutions converge strongly to the strong solution with the optimal rate 1/2.

math.NA↗

arXiv2023

Convergence analysis of the Monte Carlo method for random Navier--Stokes--Fourier system

In the present paper we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier--Stokes--Fourier system. The Monte Carlo method, which is frequently used for the approximation of statistical moments, is combined with a suitable deterministic discretisation method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to a statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for the Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.

math.NA↗

arXiv2022

Error estimates of a finite volume method for the compressible Navier--Stokes--Fourier system

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier--Stokes--Fourier system. To this end we first show the local existence of a highly regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.

math.NA↗

arXiv2022

Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method

The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.

math.NA↗

arXiv2021

Penalization method for the Navier-Stokes-Fourier system

We apply the method of penalization to the Dirichlet problem for the Navier-Stokes-Fourier system governing the motion of a general viscous compressible fluid confined to a bounded Lipschitz domain. The physical domain is embedded into a large cube on which the periodic boundary conditions are imposed. The original boundary conditions are enforced through a singular friction term in the momentum equation and a heat source/sink term in the internal energy balance. The solutions of the penalized problem are shown to converge to the solution of the limit problem. Numerical experiments are performed to illustrate the efficiency of the method.

math.AP↗

arXiv2021

Error estimates of the Godunov method for the multidimensional compressible Euler system

We derive a priori error of the Godunov method for the multidimensional Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $L^2$-norm of errors in density, momentum and entropy. Under the assumption that the numerical density and energy are bounded, we obtain a convergence rate of $1/2$ for the relative energy in the $L^1$-norm. Further, under the assumption -- the total variation of numerical solution is bounded, we obtain the first order convergence rate for the relative energy in the $L^1$-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the $L^2$-norm with the convergence rate of $1/2$. The numerical results presented for Riemann problems are consistent with our theoretical analysis.

math.NA↗