生成时间: 2026年02月16日 04:51:32

重点关注论文数量: 1

了解领域论文数量: 1

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重点关注论文（完整分析）

1. 从几乎相同的初始条件出发，纳维–斯托克斯方程光滑解的非唯一性

Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

作者: Shijun Liao, Shijie Qin

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12666v1

链接: http://arxiv.org/abs/2602.12666v1

详细分析

中文标题: 从几乎相同的初始条件出发，纳维–斯托克斯方程光滑解的非唯一性

研究对象和背景

本文研究的核心对象是二维不可压纳维–斯托克斯（Navier–Stokes, NS）方程在周期性正方形区域上的Kolmogorov 流动（Kolmogorov flow），其控制方程以流函数 $\psi(x,y,t)$ 为基本变量，采用无量纲形式表示。该系统描述了在强外力驱动下产生的湍流状态，具有典型的混沌与多尺度特征。

论文明确给出的控制方程为：

$$ \frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial(\psi,\nabla^2 \psi)}{\partial(x,y)} = \frac{1}{\mathrm{Re}} \nabla^4 \psi - n_K \cos(n_K y), $$

其中：

  • $\psi(x,y,t)$ 是流函数，满足速度场 $\mathbf{u} = (\partial_y \psi, -\partial_x \psi)$；
  • $(x,y) \in [0,2\pi] \times [0,2\pi]$，即定义在周期性正方形区域上，边界条件为：
$$ \psi(x,y,t) = \psi(x\pm2\pi,y,t) = \psi(x,y\pm2\pi,t); $$
  • $\nabla^2 = \partial_{xx} + \partial_{yy}$ 是拉普拉斯算子，$\nabla^4 = \nabla^2 \nabla^2$ 是双调和算子；
  • $\displaystyle \frac{\partial(a,b)}{\partial(x,y)} = \frac{\partial a}{\partial x}\frac{\partial b}{\partial y} - \frac{\partial a}{\partial y}\frac{\partial b}{\partial x}$ 是Jacobi 行列式（即二维涡量对流项的等价表达）；
  • $n_K = 16$ 是Kolmogorov 强迫波数，对应于外力项 $-n_K \cos(n_K y)$ 的空间频率；
  • $\mathrm{Re} = 2000$ 是雷诺数，表征惯性力与粘性力之比，处于典型湍流区间；
  • 外力项 $-n_K \cos(n_K y)$ 是经典的Kolmogorov 强迫，其平均值为零，但具有确定的空间结构，可激发各向异性湍流。

该方程组本质上等价于标准 NS 方程的涡量形式：令 $\omega = \nabla^2 \psi$ 为涡量，则原方程可重写为：

$$ \frac{\partial \omega}{\partial t} + \mathbf{u} \cdot \nabla \omega = \frac{1}{\mathrm{Re}} \nabla^2 \omega + n_K^2 \sin(n_K y). $$

但作者选择流函数形式，便于高精度谱方法实现，并自然满足不可压条件 $\nabla \cdot \mathbf{u} = 0$。

研究区域为紧致的环面 $\mathbb{T}^2 = [0,2\pi]^2$，函数空间隐含要求为周期 Sobolev 空间 $H^s_\mathrm{per}(\mathbb{T}^2)$（$s \gg 2$），以保证解的光滑性与能量守恒性质。特别地，初始数据取为：

$$ \psi_1(x,y,0) = -\frac{1}{2}\big[\cos(x+y) + \cos(x-y)\big], $$
$$ \psi_2(x,y,0) = \psi_1(x,y,0) + \delta \sin(x+y), \quad \delta = 10^{-10},\, 10^{-20},\, 10^{-40}. $$

这两个初始条件在 $L^\infty$ 范数意义下满足：

$$ \|\psi_1(\cdot,0) - \psi_2(\cdot,0)\|_{L^\infty} \leq |\delta|. $$

因此，它们构成一组任意接近的初始数据（$\delta \to 0$）。值得注意的是，$\psi_1$ 具有双重对称性：

  • 旋转对称性：$\psi_1(x,y,t) = \psi_1(2\pi - x, 2\pi - y, t)$；
  • 平移对称性：$\psi_1(x,y,t) = \psi_1(x+\pi, y+\pi, t)$；

而 $\psi_2$ 仅保留平移对称性，因扰动项 $\delta \sin(x+y)$ 在旋转下变号，破坏旋转不变性。这一微小差异成为后续解演化分岔的关键种子。

本文的研究动机根植于千禧年大奖难题之一：NS 方程在三维（或二维）中是否存在全局光滑解？其解是否唯一？ 尽管二维情形已由 Ladyzhenskaya、Kato 等人证明存在唯一全局光滑解（对任意初值 $\psi_0 \in H^1$），但该理论结果依赖于严格的数学假设（如初值属于某 Sobolev 空间、解在整个时间轴上连续可微等）。而实际物理/数值中，“光滑”与“唯一”是否在可计算、可观测、可重复的意义上成立，仍属开放问题。

尤其关键的是：传统直接数值模拟（DNS） 因固有舍入误差与截断误差，在长时间积分中必然引入 $\mathcal{O}(10^{-15})$ 量级的数值噪声；而混沌系统（如湍流）具有指数敏感依赖性（butterfly effect），导致微小误差在有限时间内放大至宏观尺度。因此，DNS 结果无法区分“真实解分歧”与“数值伪分歧”。

为此，作者引入清洁数值模拟（Clean Numerical Simulation, CNS） ——一种通过任意精度算术（multiple-precision arithmetic） 与高阶泰勒时间推进法，将总误差（截断+舍入）严格控制在预设阈值（如 $10^{-100}$）以下的新型数值范式。CNS 不是近似算法，而是一种可控误差的确定性计算框架，可在有限时间窗口 $t \in [0,T_c]$ 内生成可验证收敛的基准解，逼近真实解的程度远超传统 DNS。

综上，本文并非试图推翻二维 NS 解的存在唯一性定理，而是从计算可实现性、物理可观测性与初值敏感性角度，提出一个深刻的反例型问题：当初始条件在任意小但非零的扰动下（$\delta = 10^{-40}$），NS 湍流的长期演化是否可能产生本质不同的全局光滑轨迹？这触及了“数学唯一性”与“物理可分辨唯一性”之间的鸿沟。

主要定理或主要结果

本文未给出严格意义上的解析定理（Theorem/Proposition），因其核心结论基于高精度数值实验而非解析证明。但作者明确提出并验证了以下数值猜想（Numerical Conjecture），并视其为对 NS 唯一性问题的重要启示：

Conjecture (Non-uniqueness from arbitrarily close initial data)
设 $U_1(\mathbf{r})$ 和 $U_2(\mathbf{r}) = U_1(\mathbf{r}) + \delta U_0(\mathbf{r})$ 是二维不可压 NS 方程（在 $\mathbb{T}^2$ 上，带 Kolmogorov 强迫）的两个初始流函数，满足

$$ \|U_1 - U_2\|_{L^\infty} \leq \delta, $$


其中 $\delta > 0$ 可任意小（例如 $\delta = 10^{-40}$），且 $U_1$ 具有某种空间对称性（如旋转+平移），而 $U_2$ 仅保留部分对称性。

则存在 $T_c > 0$（称为临界可预报时间），使得对于所有 $t \in (T_c, 300]$，对应的两个 CNS 解 $\psi_1(x,y,t)$ 与 $\psi_2(x,y,t)$ 满足：

  • （i）轨迹分离：$\|\psi_1(\cdot,t) - \psi_2(\cdot,t)\|_{L^2} \geq \mathcal{O}(1)$；
  • （ii）对称性破缺：$\psi_1$ 保持初始旋转对称性，而 $\psi_2$ 完全丧失该对称性；
  • （iii）统计非等价：其动能耗散率 $D = \frac{1}{\mathrm{Re}}|\nabla \mathbf{u}|^2$ 的概率密度函数（PDF）与空间-时间平均 $\langle D\rangle_{x,t}$ 显著不同；
  • （iv）有限能量：两个解在整个 $[0,300]$ 内均满足 $\int_{\mathbb{T}^2} |\nabla \psi|^2 \, dxdy < \infty$，即具有有限动能。

该猜想由四组 CNS 实验共同支撑：Flow CNS（对应 $\psi_1$）、Flow CNS′（$\delta=10^{-10}$）、Flow CNS′（$\delta=10^{-20}$）、Flow CNS′（$\delta=10^{-40}$）。图1显示，三者在 $t \approx 100$ 后与基准解显著分离；图2证实对称性彻底丢失；图3表明统计量不可约化为同一分布。因此，该猜想实质上构成了一个数值反例族，挑战了“初值微扰不改变解本质”的经典直觉。

研究方法、关键技术和核心工具

a) 核心分析学工具

本文未使用传统偏微分方程理论中的解析工具（如不动点定理、紧性嵌入、Strichartz 估计等），而是构建了一套面向混沌系统的可验证数值分析框架。其逻辑基础在于：

  • 混沌系统的轨道分离判定准则：若两个初值距离 $\|\psi_1^0 - \psi_2^0\| < \varepsilon$，而对应解在时间 $t$ 满足 $\|\psi_1(t) - \psi_2(t)\| > \Delta$（$\Delta$ 为宏观尺度，如 $0.1$ 倍特征速度），则称发生有效轨道分离。该判据源于动力系统理论中的 Lyapunov 指数谱概念，但此处无需计算指数，仅需可观测分离。

  • 数值解的“收敛性验证”机制：CNS 的核心不是单次计算，而是双重精度验证。对同一初值，分别用两组参数 $(M_1,N_1)$ 与 $(M_2,N_2)$（$M_2>M_1$, $N_2>N_1$）运行 CNS，若二者在 $[0,T_c]$ 内差值小于 $10^{-80}$，则认定前者误差可忽略。这等价于在数值层面实现了Cauchy 序列收敛性检验，是传统 DNS 所不具备的。

  • 对称性作为解的“指纹”：利用群论中对称性破缺（symmetry breaking） 的概念，将旋转对称性的存否作为解是否“相同”的硬性判据。数学上，若解 $\psi(x,y,t)$ 满足 $\psi(x,y,t) = \psi(2\pi-x,2\pi-y,t)$ 对所有 $t$ 成立，则其属于旋转群 $C_2$ 的不变子空间；一旦该等式在某个 $t$ 失效（如图2所示），则解已跃迁至另一动力学分支。

b) 技术性难题的解决技巧

  1. 超高精度时空离散：

    • 空间离散：采用 $1024 \times 1024$ 网格的傅里叶拟谱法，将非线性项 $\partial(\psi,\nabla^2 \psi)/\partial(x,y)$ 在谱空间精确计算，避免伪谱混叠；
    • 时间推进：摒弃龙格–库塔等通用格式，改用**$M$ 阶泰勒展开**：
      $$ \psi(t+\Delta t) = \sum_{k=0}^{M} \frac{\Delta t^k}{k!} \frac{d^k \psi}{dt^k}(t), $$

      其中各阶导数由方程递推获得。当 $M=140$ 时，$\Delta t = 10^{-3}$ 下截断误差 $\sim 10^{-200}$，远低于舍入误差。

  2. 任意精度算术实现：
    所有变量（$\psi$, $\nabla^2 \psi$, 时间步长等）均以 $N_s$ 位有效数字（如 $N_s = 260$）存储与运算，借助开源库 MPFR 实现。此时舍入误差被压制至 $\sim 10^{-N_s}$，确保数值噪声不干扰物理演化。

  3. 临界可预报时间 $T_c$ 的自适应判定：
    定义 $T_c$ 为两个 CNS 解（不同 $\delta$）的 $L^2$ 距离首次超过阈值 $\eta = 10^{-2}$ 的时刻。通过扫描 $\delta = 10^{-10} \to 10^{-40}$，发现 $T_c$ 随 $\log_{10} \delta$ 近似线性增长（符合混沌系统特征时间尺度 $\sim |\log \delta| / \lambda$，$\lambda$ 为最大 Lyapunov 指数），证实分离机制普适。

  4. 统计量鲁棒性验证：
    对耗散率 $D(x,y,t)$ 计算其 PDF（直方图）与切片平均 $\langle D\rangle_x(y,t)$，并验证：同一 $\delta$ 下多次独立 CNS 运行结果重合，而不同 $\delta$ 之间 PDF 形状（峰度、偏度）与 $\langle D\rangle_x$ 的空间结构（如条纹方向）截然不同——排除随机误差影响。

这些技术共同构成一个闭环验证链：高精度 → 可验证收敛 → 对称性判据 → 统计不可约性 → 物理可解释性，从而将数值现象升华为对 NS 唯一性本质的深刻质疑。

与之前工作的比较

a) 突破与改进

本文工作与既有研究形成鲜明对比，主要体现在三个维度：

第一，初值正则性与能量约束的实质性突破。
Coiculescu & Palasek（2025）[2] 构造出非唯一解，但其初值属于临界空间 $BMO^{-1}$，且能量无限（$\|\nabla \psi_0\|_{L^2} = \infty$），物理意义薄弱。而本文初值 $\psi_1, \psi_2$ 属于 $C^\infty(\mathbb{T}^2)$，且动能 $\int |\nabla \psi_0|^2 < \infty$，完全满足物理可实现性与经典适定性理论的前提。这是首次在有限能量、任意光滑、周期边界框架下，展示非唯一性数值证据。

第二，从“存在性构造”到“可计算反例”的范式转变。
以往非唯一性工作（如 [3] 中的 dyadic 模型）依赖高度简化的代数结构，难以映射到真实 NS 方程。本文直接求解原始 NS 方程（流函数形式），所有参数（$\mathrm{Re}=2000$, $n_K=16$）均具物理意义，结果可被独立复现（代码开源），实现了从抽象存在性到具体可观察现象的跨越。

第三，揭示“混沌放大”作为非唯一性物理机制。
论文首次将 NS 湍流的噪声扩张级联（noise-expansion cascade） [16] 与初值敏感性定量关联：微小扰动 $\delta \sin(x+y)$ 并非均匀放大，而是按尺度逐级激发（从亚网格到积分尺度），最终摧毁全局对称性。这为“为何二维理论唯一性不排斥物理非唯一性”提供了机制解释——数学解空间虽唯一，但物理可观测解在混沌背景下呈现多重吸引子共存。

综上，本文不是对经典定理的否定，而是架起一座桥梁：连接解析理论的严格性与物理世界的复杂性，迫使学界重新审视“唯一性”在计算科学、实验物理与数学证明三个语境下的不同内涵。

了解领域论文（摘要翻译）

1. KdV系统快速控制代价的最优估计

Optimal bounds for the cost of fast controls of a KdV system

作者: Hoai-Minh Nguyen

类别: math.OC, math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12698v1

链接: http://arxiv.org/abs/2602.12698v1

摘要翻译

中文标题: KdV系统快速控制代价的最优估计

摘要翻译: 本文针对线性化KdV系统及局部范围内的非线性KdV系统，研究其在非临界长度情形下采用右端Neumann边界控制时的快速控制代价。由于与线性化系统相关的算子既非自伴亦非斜自伴，其（已知的）谱性质无法直接适用于矩方法，因而该问题的最优代价估计长期以来悬而未决。为克服这一困难，我们转而考察一个相关联的KdV系统，并由此导出原问题的最优代价估计。

不相关论文（基本信息）

1. 非线性扩散及其在光滑度量测度空间上的几何与函数不等式

Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces

作者: Ali Taheri

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.13025v1

链接: http://arxiv.org/abs/2602.13025v1

摘要: This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast types and their links with some geometric and functional inequalities in the framework of smooth metric measure spaces. The article presents some introduction in a summer school style as well as several new results.

2. Coulomb-Sobolev空间中薛定谔-泊松-斯莱特方程的新解

New solutions to Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

作者: Artur Jorge Marinho, Carlo Mercuri, Kanishka Perera

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12784v1

链接: http://arxiv.org/abs/2602.12784v1

摘要: We prove existence and multiplicity results for the nonlinear and nonlocal PDE

$$ - Δu + (I_α\star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$

where $N \geq 2$, $I_α: \mathbb{R}^N \setminus \{0\} \rightarrow \mathbb{R}$ is the Riesz potential of order $α\in (1,N),$ $p>1,$ and the local nonlinearity $f: [0,\infty) \times \mathbb{R} \rightarrow \mathbb R$ is subject to a new class of assumptions. We find solutions to this zero-mass problem in a Coulomb-Sobolev space using a new scaling based approach in critical point theory, by which we classify the possibly different behaviour of the nonlinearity $f$ at zero and at infinity in terms of the scaling properties of the left hand side of the equation. This is accomplished identifying a scaling invariant PDE which can be interpreted as a nonlinear eigenvalue problem, for which a sequence of eigenvalues $\{λ_k\}$ is conveniently defined via the ${\mathbb{Z}}_2$-cohomological index of Fadell and Rabinowitz. This index allows us to use new critical group estimates (and scaling-based linking sets) which might not be possible via the classical genus. Within a fairly broad set of parameters $N,α, p$ and class of assumptions on the local nonlinearity $f,$ we establish compactness results for an associated action functional and find multiple solutions as critical points, whose existence and number is sensitive to the ‘‘resonance’’ of $f$ with the sequence of eigenvalues for the scaling invariant problem, a construction which is at places reminiscent, in the present nonlinear setting, of the classical Fredholm alternative. As a byproduct of our analysis, letting $p\neq 2$ allows us to capture general nonlinearities $f$ of Sobolev-subcritical, critical, or supercritical growth.

3. 拟线性抛物型方程的定量稳定性

Quantitative stability for quasilinear parabolic equations

作者: Tapio Kurkinen, Qing Liu

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12657v1

链接: http://arxiv.org/abs/2602.12657v1

摘要: We examine the stability of a class of quasilinear parabolic partial differential equations under perturbations. We are interested in the behavior of viscosity solutions as the perturbation parameter vanishes and establish explicit convergence rates by adapting standard comparison arguments. Despite the possible singular or degenerate nature of the parabolic operator, our framework covers, in particular, both the normalized and the variational $p$-parabolic equations, providing quantitative estimates for perturbations of the exponent $p$ and limits arising from regularized approximations.

4. 在接触角条件与无通量边界条件下多维曲面扩散流方程的渐近自相似图状解

Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

作者: Yoshikazu Giga, Sho Katayama

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12619v1

链接: http://arxiv.org/abs/2602.12619v1

摘要: This paper studies Mullins’ model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.

5. 无缺陷测度情形下硬势非截断玻尔兹曼方程的整体重整化解

Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure

作者: Yi-Long Luo, Jing-Xin Nie

类别: math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12511v1

链接: http://arxiv.org/abs/2602.12511v1

摘要: The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials ($0 \leq γ\leq 1$). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ($-3 < γ< 0$).

6. 利用六边形与三角形对称性证明二维Swift–Hohenberg偏微分方程的周期解及其解支

Proving periodic solutions and branches in the 2D Swift Hohenberg PDE with hexagonal and triangular symmetry

作者: Dominic Blanco

类别: math.NA, math.AP

发布日期: 2026-02-13

arXiv ID: 2602.12491v1

链接: http://arxiv.org/abs/2602.12491v1

摘要: In this article, we enforce space group symmetries in Fourier series to rigorously prove the existence of smooth, periodic solutions in partial differential equations (PDEs) with hexagonal and triangular symmetries. In particular, we provide the necessary analytical and numerical tools to construct Fourier series of functions on the hexagonal lattice. This allows one to build approximate solutions that are periodic. Moreover, to generate the periodic tiling, we can use one symmetric hexagon for $D_6$ symmetry and two symmetric triangles for $D_3$ symmetry. We derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $\overline{u}$. More specifically, we verify a condition based on the computation of explicit bounds. The strategy for constructing $\overline{u}$, the approximate inverse, and the computation of these bounds will be presented. We demonstrate our approach on the 2D Swift-Hohenberg PDE by proving the existence of $D_3$ and $D_6$ periodic solutions. We then perform proofs of branches of solutions by using Chebyshev series. The algorithmic details to perform the proof can be found on Github.

7. 二次曲面Radon变换的微局部分析

Microlocal analysis of Radon transforms over quadric surfaces

作者: Gaik Ambartsoumian, Raluca Felea, Venkateswaran P. Krishnan, Clifford J. Nolan, Eric Todd Quinto

类别: math.CA, math.AP

发布日期: 2026-02-12

arXiv ID: 2602.12453v1

链接: http://arxiv.org/abs/2602.12453v1

摘要: We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.

8. 基于多子区域的Dirichlet–Neumann波形松弛法求解具时滞的反应-扩散方程

Dirichlet-Neumann Waveform Relaxation Method with Multiple Subdomains for Reaction-Diffusion Equation with a Time Delay

作者: Bankim C. Mandal, Deeksha Tomer

类别: math.NA, math.AP

发布日期: 2026-02-12

arXiv ID: 2602.12409v1

链接: http://arxiv.org/abs/2602.12409v1

摘要: In this study, we present the numerical investigation of the Dirichlet-Neumann Waveform Relaxation (DNWR) algorithm applied to multiple subdomains for the reaction-diffusion equation with time delay. Various arrangements of transmission conditions between subdomains are explored and a series of numerical experiments are conducted to evaluate and compare the efficiency and effectiveness of these configurations.

9. 求解尼伦伯格问题的一种机器学习方法

A Machine Learning Approach to the Nirenberg Problem

作者: Gianfranco Cortés, Maria Esteban-Casadevall, Yueqing Feng, Jonas Henkel, Edward Hirst, Tancredi Schettini Gherardini, Alexander G. Stapleton

类别: cs.LG, hep-th, math.AP, math.DG

发布日期: 2026-02-12

arXiv ID: 2602.12368v1

链接: http://arxiv.org/abs/2602.12368v1

摘要: This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on $S^2$ for metrics that are pointwise conformal to the round metric. Our mesh-free physics-informed neural network (PINN) approach directly parametrises the conformal factor globally and is trained with a geometry-aware loss enforcing the curvature equation. Additional consistency checks were performed via the Gauss-Bonnet theorem, and spherical-harmonic expansions were fit to the learnt models to provide interpretability. For prescribed curvatures with known realisability, the neural network achieves very low losses ($10^{-7} - 10^{-10}$), while unrealisable curvatures yield significantly higher losses. This distinction enables the assessment of unknown cases, separating likely realisable functions from non-realisable ones. The current capabilities of the Nirenberg Neural Network demonstrate that neural solvers can serve as exploratory tools in geometric analysis, offering a quantitative computational perspective on longstanding existence questions.