Robust physics discovery via supervised and unsupervised pattern recognition using the Euler characteristic

Machine learning approaches have been widely used for discovering the underlying physics of dynamical systems from measured data. Existing approaches, however, still lack robustness, especially when the measured data contain a large level of noise. The lack of robustness is mainly attributed to the insufficient representativeness of used features. As a result, the intrinsic mechanism governing the observed system cannot be accurately identified. In this study, we use an efficient topological descriptor for complex data, i.e., the Euler characteristics (ECs), as features to characterize the spatiotemporal data collected from dynamical systems and discover the underlying physics. Unsupervised manifold learning and supervised classification results show that EC can be used to efficiently distinguish systems with different while similar governing models. We also demonstrate that the machine learning approaches using EC can improve the confidence level of sparse regression methods of physics discovery. Introduction Scientific discovery has attracted increasing attention over the past years due to the boosting interest in exploring the new world. Data-driven approaches (machine learning, etc.) recently became possible and popular owing to revolutionary advances in sensing technologies and computational powers. Among all methods investigated for discovering scientific laws, sparse regression1 gains the most attention in extracting scientific laws (expressed as ODEs or PDEs) from measured/simulated data. Sparse regression methods inherently promote the model parsimony which is widely recognized as an intrinsic property of the governing model for a real physical system. Moreover, techniques including genetic algorithm2, reinforcement learning3, and symbolic regression4, 5 have been adopted to help build/enrich the term library of the underlying model. Sparse regression methods, however, lack robustness due to issues including inaccurate numerical differentiation (especially with large noise)6, hard-thresholding7, and incomplete library (even with library enriching techniques)2. Therefore, a robust method of scientific discovery is necessitated for discovering the underlying physics from measured data that are usually incomplete and/or imprecise. Recently, fusing prior domain knowledge in solving computational physics problems has been widely recognized as potential and promising for improving the stability, generality, and robustness of the computational model. This potential has been demonstrated in physics informed machine learning8–10 and hybrid modeling11, 12. The prior knowledge about the general physical principles has also been fused into discovering the governing PDE(s) when establishing the candidate model library5, 6. In ref.6, a machine learning classification method was proposed to detect the presence of physical patterns (such as diffusion and convection) in the investigated system and subsequently identify the governing PDE(s). Features used for this classification problem are extracted from the spatial and temporal derivatives of data and their statistics and frequency domain characteristics, considering the behaviors of physical patterns potentially existing in the investigated system. A classifier is first trained using a library of candidate models and then tested on the target system. Results show that the robustness of derived features is very limited when the level of noise goes above 0.1%, despite the novelty of this scientific discovery scheme. In this study, we use the Euler characteristics (ECs) as features for characterizing dynamical systems and will examine its robustness when large noise exists in the measured/simulated data. EC is a general topological descriptor that characterizes the geometrical features of complex datasets with reduced dimensionality and complexity while preserving the maximum information13. Compared with other topological descriptors, EC has improved generality in characterizing a wide range of mathematical objects due to its fundamental connections with other descriptive tools such as statistics, field theory, and graph theory. Given a data object, its topology will be reduced to an EC curve by using the filtration transformation technique. The general definition of the EC (χ) for a 2D manifold is: χ = β0 − β1, in which β0 is the number of 0-dimensional topological bases (known as connected components), β1 is the number of 1-dimensional topological bases (known as holes), ar X iv :2 11 0. 13 61 0v 1 [ cs .C E ] 1 5 O ct 2 02 1 and β0,β1 ∈ Z+ where Z+ is the set of nonnegative integers. The relationship between the number of topological bases and dimensions can be extended to shapes with higher dimensions. The EC of an (n+ 1)-dimensional shape is defined as χ = ∑i=0(−1)βi, in which βn ∈ Z+ is the nth Betti number and denotes the number of unique n-dimensional topological bases for the given shape14. + ×4 c = # connected components # holes (a) an example 2D shape (b) EC(c) Figure 1. Illustration on the calculation of EC for a 2D manifold. (a) A 2D shape with two connected components (i.e., β0 = 2) and four holes (i.e., β1 = 4). (b) The EC is an alternating sum of the number of connected components and holes (i.e., χ =−2). For a dynamical system, its states (or solutions, i.e., u or u) lie in a high-dimensional field Dx×Dt , in which Dx is the spatial domain and Dt is the temporal domain. As a result, an n-dimensional manifold M will be formulated with the solutions using the chart (X , f ). Here n, the dimension of the topological space, is larger than the dimension of Dx by two due to the addition of the temporal coordinate and the function f ; X ⊆ Rn−1 is a Euclidean space formed by X = Dx×Dt ; the filed/function f : X → R maps from X to a scalar function of the system solutions. Taking the dissipative system characterized by the 1D Burgers equation ut =−uux +νuxx (ν is the diffusion coefficient) as an example, its solution u(x, t) lies in a 2D field and is embedded in a 3D manifold given by a 3D chart (i.e., n = 3) (see Figure 2 (a)). The function f maps from the spatiotemporal coordinates (i.e., x and t) to the scalar solutions (i.e., u) of the equation. The chart (X , f ) is also often referred to as the graph of field f . It should be noted that the graph does not live in a Euclidean space while X does. As a result, the ECs and associated EC curve focus on the global topology of the graph during a filtration instead of the specific local connectivity information. Filtration can be applied to manifolds by virtue of the concept of superlevel set15. Given a manifold M with field f : X → R and domain X ⊆Rn, the superlevel set X` at a threshold ` ∈R is defined as: X` = {x ∈ X : f (x)> `}. The super level set has an associated field graph G` := G(X`, f`) with f` defined over X`. The field graph G` = (X`, f`) contains all points of the manifold M that have a function value greater than or equal to ` (it is a filtration of the manifold). The filtration creates a nested set of field graphs which are obtained by defining a sequence of decreasing filtration values `1 > `2 > ... > `m with associated field graphs: G`1 ⊆ G`2 ⊆ ... ⊆ G`m ⊆ G. Here, the field graphs are sparser with larger threshold values; we also have that G`m = G(X , f ) (the original graph) if `m = minx∈X f (x). For each superlevel set we obtain the field graph G` and we compute and record its EC value χ` (e.g., we determine the number of connected components and number of holes). This information is used to construct an EC curve, which contains the pairs (`,χ`). It is important to highlight that the EC curve provides a topological descriptor for a general n-dimensional field. The types of topological bases change with dimension. Taking the dissipative system characterized by the 1D Burgers equation as an example, with the solutions of the 1D Burgers equation that are embedded in a 3D manifold, the threshold in filtration is a 2D plane that cuts through the 3D graph yielding connected components and holes (see Figure 2 (a)). When the plane passes through a local maximum we have that connected components are formed, when it passes a saddle point components are joined to form holes, and when a local minimum is passed holes are filled. This reveals that filtration captures the incidence of different types of critical points in the field (its topological features) and this information is summarized in the EC curve (see Figure 2 (b)). Moreover, we found that the EC curve is immune from noise contamination (see Figure 3), which has been mathematically proved in ref.16. This advantageous property of ECs over other non-topological features will largely improve the robustness of physics discovery especially when the measured data contain a large level of noise. Therefore, this methodology we propose in this study will potentially solve the most critical challenge in data-driven physics discovery1, 5, 7. Given the measured data from an instrumented system, we formulate the manifold M on which the filtration will be