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告知研究人员以前未发现的机制，在生物系统中，跨时间和空间尺度。目前，许多这样的系统建模与偏微分方程（PDE）。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。受方程发现方法的最新进展的启发，该项目将专注于扩展方法，如生物信息神经网络（Lagergren，Nardini，Baker，Simpson，&弗洛雷斯，2020）和其他计算方法，以获得数据驱动的PDE模型。这样做，我们希望有助于模型验证的问题。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导致新的生物学见解，例如细胞损伤在增殖测定中的可能作用（Lagergren，Nardini，Baker，Simpson，&弗洛雷斯，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（布朗宁，沃恩，伯拉格，贝克和辛普森，2020年），这是至关重要的实际重要性，以获得可靠的估计，从现有数据。此外，在校准模型时，量化模型参数（以及模型预测）的不确定性非常重要。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建模选择或不应排除其他选择的有力证据。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，这一点至关重要。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）。是否可以从数据中获得可靠的估计值对模型的预测能力和可以获得的机械洞察力都有重要影响（Simpson，Baker，Vittadello，& Maclaren，2020）。开发一种方法来量化方程发现方法中函数的实际可识别性是非常重要的。这些见解可以帮助设计实验，产生区分竞争模型所需的数据。一个主要的目的是使用上述方法来估计参数和校准模型在几个应用的问题，有大量的可用数据。初步调查将关注伤口愈合试验的siRNA筛选，其目的是推断基因扰动和表型之间的联系。本研究主要包含在EPSRC的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的主题一般的连接。该项目还涉及EPSRC的人工智能技术的研究领域，因为它致力于神经网络预测的不确定性量化，统计学和应用概率，因为我们调查随机模型，最后是非线性系统，因为调查中的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.