系统生物学和系统尤其是神经生理学,最近已成为生物医学科学中许多关键应用的强大工具。然而,这样的模型通常基于需要临时计算策略并提出极高计算需求的多尺度(可能是多物理)策略的复杂组合。深度神经网络领域的最新发展证明了与传统模型相比,具有非线性,通用近似值的可能性,以估算具有高速度和准确性优势的高度非线性和复杂问题。合成数据验证后,我们使用所谓的物理约束神经网络(PINN)同时求解生物学上合理的Hodgkin-Huxley模型,并从可变和恒定电流刺激下从真实数据中推断出其参数和隐藏的时间巡回赛,显示出极低的刺激峰值和忠实信号重建的可变性。我们获得的参数范围也与先验知识兼容。我们证明可以向神经网络提供详细的生物学知识,从而使其能够在模拟和真实数据上拟合复杂的动态。
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Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.
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深度学习方法的应用加快了挑战性电流问题的分辨率,最近显示出令人鼓舞的结果。但是,电力系统动力学不是快照,稳态操作。必须考虑这些动力学,以确保这些模型提供的最佳解决方案遵守实用的动力约束,避免频率波动和网格不稳定性。不幸的是,由于其高计算成本,基于普通或部分微分方程的动态系统模型通常不适合在控制或状态估计中直接应用。为了应对这些挑战,本文介绍了一种机器学习方法,以近乎实时近似电力系统动态的行为。该拟议的框架基于梯度增强的物理知识的神经网络(GPINNS),并编码有关电源系统的基本物理定律。拟议的GPINN的关键特征是它的训练能力而无需生成昂贵的培训数据。该论文说明了在单机无限总线系统中提出的方法在预测转子角度和频率的前进和反向问题中的潜力,以及不确定的参数,例如惯性和阻尼,以展示其在一系列电力系统应用中的潜力。
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物理信息的神经网络(PINN)是神经网络(NNS),它们作为神经网络本身的组成部分编码模型方程,例如部分微分方程(PDE)。如今,PINN是用于求解PDE,分数方程,积分分化方程和随机PDE的。这种新颖的方法已成为一个多任务学习框架,在该框架中,NN必须在减少PDE残差的同时拟合观察到的数据。本文对PINNS的文献进行了全面的综述:虽然该研究的主要目标是表征这些网络及其相关的优势和缺点。该综述还试图将出版物纳入更广泛的基于搭配的物理知识的神经网络,这些神经网络构成了香草·皮恩(Vanilla Pinn)以及许多其他变体,例如物理受限的神经网络(PCNN),各种HP-VPINN,变量HP-VPINN,VPINN,VPINN,变体。和保守的Pinn(CPINN)。该研究表明,大多数研究都集中在通过不同的激活功能,梯度优化技术,神经网络结构和损耗功能结构来定制PINN。尽管使用PINN的应用范围广泛,但通过证明其在某些情况下比有限元方法(FEM)等经典数值技术更可行的能力,但仍有可能的进步,最著名的是尚未解决的理论问题。
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物理知识的神经网络(PINNS)最近由于解决前进和反向问题的能力而受到了很多关注。为了训练与PINN相关的深层神经网络,通常会使用不同损失项的加权总和构建总损耗函数,然后尝试将其最小化。这种方法通常会成为解决刚性方程式的问题,因为它不能考虑自适应增量。许多研究报告说,PINN的性能不佳及其在模拟僵硬的普通差分条件(ODE)条件下模拟僵硬的化学活动问题方面的挑战。研究表明,刚度是PINN在模拟刚性动力学系统中失败的主要原因。在这里,我们通过提出减少损失函数的弱形式来解决这个问题,这导致了新的PINN结构(进一步称为还原Pinn),该结构利用降低的集成方法来使Pinn能够求解僵硬的化学动力学。所提出的还原细菌可以应用于涉及僵硬动力学的各种反应扩散系统。为此,我们将初始价值问题(IVP)转换为它们的等效积分形式,并使用物理知识的神经网络求解所得的积分方程。在我们派生的基于积分的优化过程中,只有一个术语,而没有明确合并与普通微分方程(ODE)和初始条件(ICS)相关的损失项。为了说明减少细菌的功能,我们用它来模拟多个僵硬/轻度的二阶频率。我们表明,还原的Pinn可准确捕获刚性标量颂歌的溶液。我们还针对线性ODES的硬质系统验证了还原的Pinn。
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科学和工程学中的一个基本问题是设计最佳的控制政策,这些政策将给定的系统转向预期的结果。这项工作提出了同时求解给定系统状态和最佳控制信号的控制物理信息的神经网络(控制PINNS),在符合基础物理定律的一个阶段框架中。先前的方法使用两个阶段的框架,该框架首先建模然后按顺序控制系统。相比之下,控制PINN将所需的最佳条件纳入其体系结构和损耗函数中。通过解决以下开环的最佳控制问题来证明控制PINN的成功:(i)一个分析问题,(ii)一维热方程,以及(iii)二维捕食者捕食者问题。
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当通过差异模型研究流行动力学时,要了解现象并模拟预测场景所需的参数需要微妙的校准阶段,通常会因官方来源报告的稀缺性和不确定性而变得更加挑战。在这种情况下,通过嵌入控制物理现象在学习过程中的差异模型的知识,可以有效解决数据驱动的学习的逆问题,并解决相应的流行病问题,从而使物理知识的神经网络(PINN)(PINN)(PINN)(PINNS)。 。然而,在许多情况下,传染病的空间传播的特征是在多尺度PDE的不同尺度上的个体运动。这反映了与城市和邻近区域内动态有关的区域或领域的异质性。在存在多个量表的情况下,PINN的直接应用通常会导致由于神经网络损失函数中差异模型的多尺度性质而导致的结果差。为了使神经网络相对于小规模统一运行,希望神经网络满足学习过程中的渐近保护(AP)特性。为此,我们考虑了一类新的AP神经网络(APNNS),用于多尺度双曲线传输模型的流行病扩散模型,由于损失函数的适当配方,它能够在系统的不同尺度上均匀地工作。一系列针对不同流行病的数值测试证实了所提出的方法的有效性,在处理多尺度问题时,突出了AP在神经网络中的重要性,尤其是在存在稀疏和部分观察到的系统的情况下。
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Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.
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Solute transport in porous media is relevant to a wide range of applications in hydrogeology, geothermal energy, underground CO2 storage, and a variety of chemical engineering systems. Due to the complexity of solute transport in heterogeneous porous media, traditional solvers require high resolution meshing and are therefore expensive computationally. This study explores the application of a mesh-free method based on deep learning to accelerate the simulation of solute transport. We employ Physics-informed Neural Networks (PiNN) to solve solute transport problems in homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that learn from large training datasets, PiNNs only leverage the strong form mathematical models to simultaneously solve for multiple dependent or independent field variables (e.g., pressure and solute concentration fields). In this study, we construct PiNN using a periodic activation function to better represent the complex physical signals (i.e., pressure) and their derivatives (i.e., velocity). Several case studies are designed with the intention of investigating the proposed PiNN's capability to handle different degrees of complexity. A manual hyperparameter tuning method is used to find the best PiNN architecture for each test case. Point-wise error and mean square error (MSE) measures are employed to assess the performance of PiNNs' predictions against the ground truth solutions obtained analytically or numerically using the finite element method. Our findings show that the predictions of PiNN are in good agreement with the ground truth solutions while reducing computational complexity and cost by, at least, three orders of magnitude.
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科学机器学习(Sciml)的出现在思路科学领域开辟了一个新的领域,通过在基于物理和数据建模的界面的界面中开发方法。为此,近年来介绍了物理知识的神经网络(Pinns),通过在所谓的焊点上纳入物理知识来应对培训数据的稀缺。在这项工作中,我们研究了Pinns关于用于强制基于物理惩罚术语的配偶数量的预测性能。我们表明Pinns可能会失败,学习通过定义来满足物理惩罚术语的琐碎解决方案。我们制定了一种替代的采样方法和新的惩罚术语,使我们能够在具有竞争性结果的数据稀缺设置中纠正Pinns中的核心问题,同时减少最多80 \%的基准问题所需的搭配数量。
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由于在许多领域的无与伦比的成功,例如计算机视觉,自然语言处理,推荐系统以及最近在模拟多物理问题和预测非线性动力学系统方面,深度学习引起了人们的关注。但是,建模和预测混乱系统的动态仍然是一个开放的研究问题,因为训练深度学习模型需要大数据,在许多情况下,这并不总是可用的。可以通过从模拟结果获得的其他信息以及执行混乱系统的物理定律来培训这样的深度学习者。本文考虑了极端事件及其动态,并提出了基于深层神经网络的优雅模型,称为基于知识的深度学习(KDL)。我们提出的KDL可以通过直接从动力学及其微分方程中对真实和模拟数据进行联合培训来学习控制混乱系统的复杂模式。这些知识被转移到模型和预测现实世界中的混乱事件,表现出极端行为。我们通过在三个实际基准数据集上进行评估来验证模型的效率:El Nino海面温度,San Juan登革热病毒感染和BJ {\ o} rn {\ o} ya每日降水,所有这些都受极端事件的控制'动态。利用对极端事件和基于物理的损失功能的先验知识来领导神经网络学习,我们即使在小型数据制度中也可以确保身体一致,可推广和准确的预测。
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地震波的频域模拟在地震反演中起着重要作用,但在大型模型中仍然具有挑战性。作为有效的深度学习方法,最近提出的物理知识的神经网络(PINN)在解决广泛的偏微分方程(PDES)方面取得了成功的应用,并且在这方面仍然有改进的余地。例如,当PDE系数不平滑并描述结构复合介质时,PINN可能导致溶液不准确。在本文中,我们通过使用PINN而不是波方程来求解频域中的声学和Visco声学散射的场波方程,以消除源奇异性。我们首先说明,当在损失函数中未实现边界条件时,非平滑速度模型导致波场不准确。然后,我们在PINN的损耗函数中添加了完美匹配的层(PML)条件,并设计了二次神经网络,以克服PINN中非平滑模型的有害影响。我们表明,PML和二次神经元改善了结果和衰减,并讨论了这种改进的原因。我们还说明,在波场模拟中训练的网络可用于预先训练PDE-Coeff及时改变后另一个波场模拟的神经网络,并相应地提高收敛速度。当两次连续迭代或两个连续的实验之间的模型扰动时,这种预训练策略应在迭代全波形反转(FWI)和时置目标成像中找到应用。
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本文侧重于各种技术来查找替代近似方法,可以普遍用于各种CFD问题,但计算成本低,运行时低。在机器学习领域中探讨了各种技术,以衡量实现核心野心的效用。稳定的平流扩散问题已被用作测试用例,以了解方法可以提供解决方案的复杂程度。最终,该重点留在物理知识的机器学习技术上,其中求解微分方程是可能的,而无需计算数据。 i.e的普遍方法拉加里斯et.al.和M. Raissi et.al彻底探讨。普遍存在的方法无法解决占主导地位问题。提出了一种称为分布物理知识神经网络(DPINN)的物理知情方法,以解决平流的主导问题。它通过分割域并将其他基于物理的限制引入均方平方损耗条款来增加旧方法的可执行和能力。完成各种实验以探索结束与该方法结束的最终可能性。也完成了参数研究以了解方法对不同可调参数的方法。该方法经过稳定的平流 - 扩散问题和不稳定的方脉冲问题。记录非常准确的结果。极端学习机(ELM)是一种以可调谐参数成本的快速神经网络算法。在平面扩散问题上测试所提出的模型的基于ELM的变体。榆树使得复杂优化更简单,并且由于该方法是非迭代的,因此解决方案被记录在单一镜头中。基于ELM的变体似乎比简单的DPINN方法更好。在本文中,将来同时进行各种发展的范围。
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The identification of material parameters occurring in constitutive models has a wide range of applications in practice. One of these applications is the monitoring and assessment of the actual condition of infrastructure buildings, as the material parameters directly reflect the resistance of the structures to external impacts. Physics-informed neural networks (PINNs) have recently emerged as a suitable method for solving inverse problems. The advantages of this method are a straightforward inclusion of observation data. Unlike grid-based methods, such as the finite element method updating (FEMU) approach, no computational grid and no interpolation of the data is required. In the current work, we aim to further develop PINNs towards the calibration of the linear-elastic constitutive model from full-field displacement and global force data in a realistic regime. We show that normalization and conditioning of the optimization problem play a crucial role in this process. Therefore, among others, we identify the material parameters for initial estimates and balance the individual terms in the loss function. In order to reduce the dependence of the identified material parameters on local errors in the displacement approximation, we base the identification not on the stress boundary conditions but instead on the global balance of internal and external work. In addition, we found that we get a better posed inverse problem if we reformulate it in terms of bulk and shear modulus instead of Young's modulus and Poisson's ratio. We demonstrate that the enhanced PINNs are capable of identifying material parameters from both experimental one-dimensional data and synthetic full-field displacement data in a realistic regime. Since displacement data measured by, e.g., a digital image correlation (DIC) system is noisy, we additionally investigate the robustness of the method to different levels of noise.
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Non-equilibrium chemistry is a key process in the study of the InterStellar Medium (ISM), in particular the formation of molecular clouds and thus stars. However, computationally it is among the most difficult tasks to include in astrophysical simulations, because of the typically high (>40) number of reactions, the short evolutionary timescales (about $10^4$ times less than the ISM dynamical time) and the characteristic non-linearity and stiffness of the associated Ordinary Differential Equations system (ODEs). In this proof of concept work, we show that Physics Informed Neural Networks (PINN) are a viable alternative to traditional ODE time integrators for stiff thermo-chemical systems, i.e. up to molecular hydrogen formation (9 species and 46 reactions). Testing different chemical networks in a wide range of densities ($-2< \log n/{\rm cm}^{-3}< 3$) and temperatures ($1 < \log T/{\rm K}< 5$), we find that a basic architecture can give a comfortable convergence only for simplified chemical systems: to properly capture the sudden chemical and thermal variations a Deep Galerkin Method is needed. Once trained ($\sim 10^3$ GPUhr), the PINN well reproduces the strong non-linear nature of the solutions (errors $\lesssim 10\%$) and can give speed-ups up to a factor of $\sim 200$ with respect to traditional ODE solvers. Further, the latter have completion times that vary by about $\sim 30\%$ for different initial $n$ and $T$, while the PINN method gives negligible variations. Both the speed-up and the potential improvement in load balancing imply that PINN-powered simulations are a very palatable way to solve complex chemical calculation in astrophysical and cosmological problems.
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We propose characteristic-informed neural networks (CINN), a simple and efficient machine learning approach for solving forward and inverse problems involving hyperbolic PDEs. Like physics-informed neural networks (PINN), CINN is a meshless machine learning solver with universal approximation capabilities. Unlike PINN, which enforces a PDE softly via a multi-part loss function, CINN encodes the characteristics of the PDE in a general-purpose deep neural network trained with the usual MSE data-fitting regression loss and standard deep learning optimization methods. This leads to faster training and can avoid well-known pathologies of gradient descent optimization of multi-part PINN loss functions. If the characteristic ODEs can be solved exactly, which is true in important cases, the output of a CINN is an exact solution of the PDE, even at initialization, preventing the occurrence of non-physical outputs. Otherwise, the ODEs must be solved approximately, but the CINN is still trained only using a data-fitting loss function. The performance of CINN is assessed empirically in forward and inverse linear hyperbolic problems. These preliminary results indicate that CINN is able to improve on the accuracy of the baseline PINN, while being nearly twice as fast to train and avoiding non-physical solutions. Future extensions to hyperbolic PDE systems and nonlinear PDEs are also briefly discussed.
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微分方程在现代世界中起着关键作用,包括科学,工程,生态,经济学和金融,这些方程可用于模拟许多物理系统和过程。在本文中,我们使用物理知识的神经网络(PINN)研究了人类系统中药物同化的两个数学模型。在第一个模型中,我们考虑了人类系统中单剂量的单剂量的情况,在第二种情况下,我们考虑定期服用这种药物的过程。我们已经使用隔室图来对这些情况进行建模。使用PINN求解所得的微分方程,在该方程中,我们使用feed向前的多层感知器作为函数近似器,并且对网络参数进行调整以获取最小误差。此外,通过找到有关网络参数的误差函数的梯度来训练网络。我们采用了用于PINNS的Python库DeepXde来求解描述两种药物同化模型的一阶微分方程。结果显示,确切解决方案和预测解之间的高度准确性与第一个模型的结果误差达到10^(-11),而第二个模型的误差为10^(-8)。这验证了PINN在求解任何动态系统中的使用。
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化学动力学和反应工程包括解除反应机制的现象学框架,优化反应性能和化学过程的合理设计。这里,我们利用前馈人工神经网络作为基础函数来解决由描述微蓄电图(MKMS)的差分代数方程(DAE)约束的常微分方程(杂物)。我们提出了一种代数框架,用于反应网络,基本反应类型和化学物种的数学描述和分类。在该框架下,我们证明了在正则化的多目标优化设置中同时训练了神经网络和动力学模型参数,通过估计来自合成实验数据的动力学参数来导致逆问题的解决方案。我们分析了一组方案,以确定可以从瞬态动力学数据检索动力学参数的程度,并评估方法的鲁棒性相对于统计噪声。这种反向动力学杂散的方法可以帮助基于瞬态数据阐明反应机制。
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量子计算有望加快科学和工程中的一些最具挑战性问题。已经提出了量子算法,显示了从化学到物流优化的应用中的理论优势。科学和工程中出现的许多问题可以作为一组微分方程重写。用于求解微分方程的量子算法已经示出了容错量计算制度中的可提供的优势,其中深宽的量子电路可用于求解局部微分方程(PDES)的大型线性系统。最近,提出了求解非线性PDE的变分方法也具有近术语量子器件。最有前途的一般方法之一是基于近期科学机器学习领域的发展来解决PDE。我们将近期量子计算机的适用性扩展到更一般的科学机器学习任务,包括从测量数据集发现微分方程。我们使用可分辨率量子电路(DQC)来解决由操作员库参数化的等式,并在数据和方程的组合上执行回归。我们的结果显示了普通模型发现(QMOD)的有希望的路径,在经典和量子机器学习方法之间的界面上。我们在不同系统上展示了成功的参数推断和方程发现,包括二阶,常微分方程和非线性部分微分方程。
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Navier-Stokes方程是描述液体和空气等流体运动的重要部分微分方程。由于Navier-Stokes方程的重要性,有效的数值方案的发展对科学和工程师都很重要。最近,随着AI技术的开发,已经设计了几种方法来整合深层神经网络,以模拟和推断不可压缩的Navier-Stokes方程所控制的流体动力学,这些方程可以以无网状和可不同的方式加速模拟或推断过程。在本文中,我们指出,现有的深入Navier-Stokes知情方法的能力仅限于处理非平滑或分数方程,这在现实中是两种关键情况。为此,我们提出了\ emph {深入的随机涡流方法}(drvm),该方法将神经网络与随机涡流动力学系统相结合,等效于Navier-Stokes方程。具体而言,随机涡流动力学激发了用于训练神经网络的基于蒙特卡洛的损失函数,从而避免通过自动差异计算衍生物。因此,DRVM不仅可以有效地求解涉及粗糙路径,非差异初始条件和分数运算符的Navier-Stokes方程,而且还继承了基于深度学习的求解器的无网格和可区分优势。我们对凯奇问题,参数求解器学习以及2-D和3-D不可压缩的Navier-Stokes方程的逆问题进行实验。所提出的方法为Navier-Stokes方程的仿真和推断提供了准确的结果。特别是对于包括奇异初始条件的情况,DRVM明显胜过现有的PINN方法。
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