Computation of steady-state probabilities is an important aspect of analysing biological systems modelled as probabilistic Boolean networks (PBNs). For small PBNs, efficient numerical methods to compute steady-state probabilities of PBNs exist, based on the Markov chain state-transition matrix. However, for large PBNs, numerical methods suffer from the state-space explosion problem since the state-space size is exponential in the number of nodes in a PBN. In fact, the use of statistical methods and Monte Carlo methods remain the only feasible approach to address the problem for large PBNs. Such methods usually rely on long simulations of a PBN. Since slow simulation can impede the analysis, the efficiency of the simulation procedure becomes critical. Intuitively, parallelising the simulation process is the ideal way to accelerate the computation. Recent developments of general purpose graphics processing units (GPUs) provide possibilities to massively parallelise the simulation process. In this work, we propose a trajectory-level parallelisation framework to accelerate the computation of steady-state probabilities in large PBNs with the use of GPUs. To maximise the computation efficiency on a GPU, we develop a dynamical data arrangement mechanism for handling different size PBNs with a GPU. Specially, we propose a reorder-and-split method to handle both large and dense PBNs. Besides, we develop a specific way of storing predictor functions of a PBN and the state of the PBN in the GPU memory. Moreover, we introduce a strongly connected component (SCC)-based network reduction technique to further accelerate the computation speed. Experimental results show that our GPU-based parallelisation gains approximately a 600-fold speedup for a real-life PBN compared to the state-of-the-art sequential method.

中文翻译：

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大型概率布尔网络的 GPU 加速稳态计算

稳态概率的计算是分析被建模为概率布尔网络 (PBN) 的生物系统的一个重要方面。对于小型 PBN，基于马尔可夫链状态转移矩阵，存在计算 PBN 稳态概率的有效数值方法。然而，对于大型 PBN，数值方法会遇到状态空间爆炸问题，因为状态空间大小是 PBN 中节点数量的指数。事实上，使用统计方法和蒙特卡罗方法仍然是解决大型 PBN 问题的唯一可行方法。这种方法通常依赖于对 PBN 的长时间模拟。由于缓慢的模拟会阻碍分析，因此模拟过程的效率变得至关重要。直观地说，并行化仿真过程是加速计算的理想方式。通用图形处理单元 (GPU) 的最新发展为大规模并行化模拟过程提供了可能性。在这项工作中，我们提出了一个轨迹级并行化框架使用 GPU 加速计算大型 PBN 中的稳态概率。为了最大限度地提高 GPU 上的计算效率，我们开发了一种动态数据排列机制，用于使用 GPU 处理不同大小的 PBN。特别地，我们提出了一个重新排序和拆分处理大型和密集 PBN 的方法。此外，我们开发了一种将 PBN 的预测函数和 PBN 的状态存储在 GPU 内存中的特定方法。此外，我们引入了一种基于强连接组件（SCC）的网络缩减技术，以进一步加快计算速度。实验结果表明，与最先进的顺序方法相比，我们基于 GPU 的并行化在实际 PBN 中获得了大约 600 倍的加速。