The Physarum computing model is an analog computing model motivated by the network dynamics of the slime mold Physarum Polycephalum. In previous works, it was shown that it can solve a class of linear programs. We extend these results to a more general dynamics motivated by situations where the slime mold operates in a non-uniform environment.

Let $c\in {\mathbb{Z}}_{>0}^m$, $A\in {\mathbb{Z}}^{n\times m}$, and $b\in {\mathbb{Z}}^n$. We show under fairly general conditions that the non-uniform Physarum dynamics${\dot{x}}_e=a_e(x,t)(|q_e|-x_e)$ converges to the optimum solution $x^{⁎}$ of the weighted basis pursuit problem minimize $c^{T}x$ subject to $Af=b$ and $|f|\le x$. Here, f and x are m-dimensional vectors of real variables, q minimizes the energy ${\sum }_e(c_e/x_e)q_e^2$ subject to the constraints $Aq=b$ and $\mathrm{supp}(q)\subseteq \mathrm{supp}(x)$, and $a_e(x,t)>0$ is the reactivity of edge e to the difference $|q_e|-x_e$ at time t and in state x. Previously convergence was only shown for the uniform case $a_e(x,t)=1$ for all e, x, and t.

We also show convergence for the dynamics${\dot{x}}_e=x_e(g_e\left(\frac{|q_e|}{x_e}\right)-1),$ where each $g_e$ is an increasing differentiable function with $g_e(1)=1$ (satisfying some mild conditions). Previously, convergence was only shown for the special case of the shortest path problem on a graph consisting of two nodes connected by parallel edges.

Physarum计算模型是一种模拟计算模型，该模型是由粘液霉菌Physarum Polycephalum的网络动力学驱动的。在以前的工作中，它可以解决一类线性程序。我们将这些结果扩展到更广泛的动态范围，这些动态范围是由粘液模在非均匀环境中运行的情况引起的。

让 $C\in {\mathbb{ž}}_{>0}^{米}$， $\mathrm{一种}\in {\mathbb{ž}}^{ñ\times 米}$和 $b\in {\mathbb{ž}}^{ñ}$。我们在相当普遍的条件下证明了非均匀的Phys骨动力学${\dot{X}}_{Ë}={\mathrm{一种}}_{Ë}（X，Ť）（|q_{Ë}|-X_{Ë}）$ 收敛到最佳解决方案 $X^{⁎}$ 加权基准追踪问题的最小化 $C^{Ť}X$ 服从 $\mathrm{一种}F=b$ 和 $|F|\le X$。在这里，f和x是实变量的m维向量，q使能量最小${\sum }_{Ë}（C_{Ë}/X_{Ë}）q_{Ë}^2$ 受约束 $\mathrm{一种}q=b$ 和 $\mathrm{支持}（q）\subseteq \mathrm{支持}（X）$和 ${\mathrm{一种}}_{Ë}（X，Ť）>0$是边e对差异的反应性$|q_{Ë}|-X_{Ë}$在时间t和状态x。以前仅在统一情况下显示收敛${\mathrm{一种}}_{Ë}（X，Ť）=\mathrm{1个}$对于所有e，x和t。

我们还展示了动力学的收敛性${\dot{X}}_{Ë}=X_{Ë}（G_{Ë}（\frac{|q_{Ë}|}{X_{Ë}}）-\mathrm{1个}），$ 每个在哪里 $G_{Ë}$ 是一个越来越微分的功能 $G_{Ë}（\mathrm{1个}）=\mathrm{1个}$（满足一些温和的条件）。以前，仅在图上最短路径问题的特殊情况下才显示收敛性，该图由两个由平行边连接的节点组成。