New methods for model construction and calibration using machine learning and mathematical modelling.

越来越多的生物数据的可用性应该使研究人员了解生物系统中以前未被发现的机制，跨越时间和空间尺度。目前，许多这样的系统都是用偏微分方程（PDEs）建模的。然而，对于大多数问题，存在相互竞争的PDE模型，其中一些具有不同的解释。受方程发现方法最新进展的推动，该项目将重点扩展生物信息神经网络（Lagergren, Nardini, Baker, Simpson, & Flores, 2020）等方法和其他计算方法，以获得数据驱动的PDE模型。通过这样做，我们希望对模型验证的问题有所贡献。这在数学上和生物学上都很有趣，因为发现PDE模型中的新术语已经显示出新的生物学见解，例如细胞损伤在增殖试验中的可能作用（Lagergren, Nardini, Baker, Simpson, & Flores, 2020）。鉴于生物学中数学模型的目标范围从建立预测模型到量化无法测量的参数（Browning, Warne, Burrage, Baker, & Simpson, 2020），从现有数据中获得可靠的估计具有至关重要的实际意义。此外，在校准模型时，量化模型参数中的不确定性（因此，在模型预测中）是很重要的。不确定性量化不仅提供了关于模型可能误差的信息，而且还可以表明现有数据是否提供了支持模型选择的有力证据，或者不应排除其他选择。目前，还没有公认的方法来对方程学习方法进行这种分析，而如果要找到对潜在动力学的可靠见解，这是至关重要的。从现有数据中获得可靠估计的另一个问题是参数可识别性（Simpson, Baker, Vittadello, & Maclaren, 2020）。能否从数据中获得可靠的估计对于模型的预测能力和可以获得的机制洞察力都具有重要的影响（Simpson, Baker, Vittadello, & Maclaren, 2020）。在方程发现方法中，开发一种量化函数实际可辨识性的方法是非常重要的。这样的见解可以帮助设计实验产生必要的数据，以区分竞争模型。一个主要的目的是使用上述方法估计参数和校准模型在几个应用问题，其中有大量的可用数据。初步调查将涉及伤口愈合试验的siRNA筛选，其目的是推断基因扰动与表型之间的联系。本研究主要包含在EPSRC数学生物学研究领域内。在数学生物学中，我们指出了与医疗保健技术和数学科学主题的联系。该项目还涉及EPSRC的人工智能技术研究领域，因为它研究神经网络预测的不确定性量化，统计学和应用概率，因为我们研究随机模型，最后是非线性系统，因为所研究的PDE模型通常包含非线性。

An increasing availability of biological data should inform researchers of previously undiscovered mechanisms in biological systems, across temporal and spatial scales. Currently, many such systems are modeled with partial differential equations (PDEs). However, for most problems competing PDE models exist, some with different interpretations. Motivated by recent advances in equation discovery methods, this project will focus on extending methods such as biologically informed neural networks (Lagergren, Nardini, Baker, Simpson, & Flores, 2020) and other computational approaches to obtain data-driven PDE models. In so doing, we hope to contribute to the problem of model validation. This is of interest mathematically as well as biologically, as discovering new terms in PDE models has already shown to lead to new biological insights, such as the possible role of cell damage in proliferation assays (Lagergren, Nardini, Baker, Simpson, & Flores, 2020).Given that the goals of mathematical models in biology range from building predictive models to quantifying parameters that cannot be measured (Browning, Warne, Burrage, Baker, & Simpson, 2020), it is of vital practical importance to obtain reliable estimates from available data. Further, when calibrating models, it is important to quantify uncertainty in model parameters (and, consequently, in model predictions). Uncertainty quantification provides not only information about the possible error of the model but can also suggest whether the available data provide strong evidence in favour of the modelling choices or that other choices ought not be excluded. Currently, there is no accepted way to do this analysis for equation learning methods, while it is of vital importance if reliable insights into the underlying dynamics are to be found. Another issue in obtaining reliable estimates from available data is parameter identifiability (Simpson, Baker, Vittadello, & Maclaren, 2020). Whether reliable estimates can at all be obtained from the data has important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained (Simpson, Baker, Vittadello, & Maclaren, 2020). A means to develop a way to quantify practical identifiability of functions in equation discovery methods is hugely important. Such insights can aid design experiments yielding the data necessary to discriminate between competing models. A principal aim is to use the methods outlined above to estimate parameters and calibrate models in several applied problems where there is a large amount of available data. An initial investigation will be concerned with siRNA screens of wound healing assays where the aim would be to infer the link between gene perturbation and phenotype. This research is mainly contained within the EPSRC research area of mathematical biology. Within mathematical biology, we point out the connection with the themes of healthcare technologies and mathematical sciences in general. This project also relates to the EPSRC's research areas of Artificial Intelligence Technologies as it works on uncertainty quantification of neural network predictions, Statistics and Applied Probability as we investigate stochastic models and finally Non-Linear Systems as the PDE models under investigation often contain non-linearities.