自由能計算為分子動力學模擬一大重要且熱門的研究主題。倘若目標系統的自
由能分布能被良好的定義與計算,其對應的熱力學性質就能被精準的求出,從而促
進於該系統的應用。然而,若目標系統存在高自由能障,分子動力學所依據的遍歷
假說,將會失效。此時,採樣將不完全,所計算的熱力學參數便失去精確性,最終
模擬將無法得到收斂的自由能。為解決採樣不足的問題,進階採樣演算法相關研究,無論是基於機率分佈, 或者是力分佈, 如熱力學積分法 (Kirkwood 1935), 自由能微擾法(Zwanzig 1954), 傘型採樣 (Torrie and Valleau 1977), metadynamics (A Laio 2002), 以及 adaptive biasing force (Darve 2002) 等於過去數十年間,逐一提出。然而對複雜系統,進階採樣法還是存在採樣效率以及收斂性的問題,特別是系統邊界或高能障區域。
在本研究之中,我們利用機器學習中的類神經網路 (artificial neural network, ANN), 改善傳統進階採樣法的取樣效率與收斂性。結果顯示,搭配 ANN 的進階採樣方法, 如 adaptive biasing force (ABF), adaptive umbrella sampling (AUS) method (Mezei 1987,Bartels 1997), 統計溫度分子動力學法 (STMD, Kim 2006) 及 COLVARS 模組中 (Fiorin
2013) 的 ABF 功能,可更快、更準確的收斂到真實的自由能分佈。其中,在一維
及二維簡易模型計算中,ABF 的收斂性比 AUS 更好。另外,標竿測試部分,利用
STMD 及 COLVARS 模組計算真實液體系統,也如預期有良好表現。本研究開發的 ANN 進階取樣法的優點為: (1) 無須預知系統的自由能部分分布資訊;(2) 提供平滑且連續的自由能分布預測,甚至是當前尚未採樣到的區域;(3) 加入 ANN 訓練準則,從而降低誤差,提高採樣品質,進而提升 ANN 之預測精確度與效率;(4) 無須預先累積 ANN 訓練集或者預設目標函數 (如機率分佈),特別適合動力學特性較差的系統;(5) 能與現有分子動力學軟體做結合。基於上述優點,運用 ANN 輔佐進階取樣法能提供高效、高精確度以及低數值誤差的計算結果,自由能分布能被快速計算求出,預期可促進相關研究領域的發展。
關鍵字: 類神經網路,自由能計算,分子動力學

Free energy calculation has been an important field of molecular dynamics (MD). With the accurate free energy profile, the associated thermodynamic properties of the system can be fully identified, which allows better utilization of the materials to the real-world application.
However, it is difficult for the conventional MD algorithm to achieve full and ideal sampling, i.e., satisfying ergodic hypothesis, due to the existing high energy barriers in the system. Consequently, the simulation fails to obtain a converged free energy landscape. Hence,
various enhanced sampling methods of either histogram or force-based methods have been proposed to improve the sampling efficiency, such as thermodynamics integration (Kirkwood 1935), free energy perturbation (Zwanzig 1954), umbrella sampling (Torrie and Valleau 1977), metadynamics (A Laio 2002), and adaptive biasing force (ABF) (Darve 2002)etc. However, such enhanced sampling method might still suffer from insufficient sampling efficiency, slower convergence rate, and numerical error near system boundaries or high energy states. In this work, we adopt artificial neural network (ANN), a subset of machine
learning (ML), to further improve the conventional algorithms, including ABF, adaptive umbrella sampling (AUS, Mezei 1987, Bartels 1997), statistical temperature molecular dynamics (STMD, Kim 2006), and ABF in the COLVARS (Fiorin 2013). The results show that the ANN enhanced free energy methods are able to converge to their correct free energy landscape much faster than the original free energy algorithms. For the calculation of ABF and AUS on 1D and 2D toy model systems, ABF has better convergence rate than AUS. For the benchmark, STMD and ABF in COLVARS module coupled with ANN again shows better performance compared with the conventional method without the support of ANN. The advantages of our ANN framework include: (1) the a priori knowledge to the system free energy profile is not required; (2) smooth, unwrinkled, and continuous free energy estimates,even for the unsampled regions, is offered; (3) training criteria is set for lower error and high sampling quality, which offers better estimate; (4) the pre-trained data is not essential since it is accumulated during simulation, and pre-assumed target function (e.g. probability distribution) is not required as well. These are benefits for the calculation of the slow dynamics systems; (5) the ability to link with the common MD software. With the aforementioned advantages, our ANN-based framework can obtain a converged free energy profile of the system with better efficiency, higher accuracy, lower numerical error. With conventional MD algorithms coupled with the ANN framework, our work ultimately helps facilitate the research progress of the related fields.
Keywords:Artificial Neural Network, Free Energy Calculation, Molecular Dynamics

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