Learning the nonlinear flux function of a hidden scalar conservation law from data

Nonlinear conservation laws are widely used in fluid mechanics, biology, physics, and chemical engineering. However, deriving such nonlinear conservation laws is a significant and challenging problem. A possible attractive approach is to extract conservation laws more directly from observation data by use of machine learning methods. We propose a framework that combines a symbolic multi-layer neural network and a discrete scheme to learn the nonlinear, unknown flux function f(u) of the scalar conservation law ut + f(u)x = 0 * with u as the main variable. This identification is based on using observation data u(xj,ti) on a spatial grid xj, j = 1, ... Nx at specified times ti, i = 1, ..., Nobs. A main challenge with Eq (*) is that the solution typically creates shocks, i.e., one or several jumps of the form (uL, uR) with uL ≠ uR moving in space and possibly changing over time such that information about f(u) in the interval associated with this jump is sparse or not at all present in the observation data. Secondly, the lack of regularity in the solution of (*) and the nonlinear form of f(u) hamper use of previous proposed physics informed neural network (PINN) methods where the underlying form of the sought differential equation is accounted for in the loss function. We circumvent this obstacle by approximating the unknown conservation law (*) by an entropy satisfying discrete scheme where f(u) is represented through a symbolic multi-layer neural network. Numerical experiments show that the proposed method has the ability to uncover the hidden conservation law for a wide variety of different nonlinear flux functions, ranging from pure concave/convex to highly non-convex shapes. This is achieved by relying on a relatively sparse amount of observation data obtained in combination with a selection of different initial data.publishedVersio

Can cancer cells inform us about the tumor microenvironment?

Characteristics of the tumor microenvironment (TME) such as the leaky intratumoral vascular network and the density and composition of the desmoplastic extracellular matrix (ECM) contain essential information that determine the possibly heterogeneous interstitial fluid (IF) velocity field and interstitial fluid pressure (IFP). This information plays an important role for how anticancer drug that is delivered through the blood vasculature will distribute and possibly affect the tumor. The main question we deal with in this work is: Can we lure the cancer cells to reveal such information to us? By means of an in silico tumor model we demonstrate that subject to the condition that the tumor progression behavior is dominated by a cancer cell phenotype which moves by fluid-sensitive migration mechanisms as reported from experimental works, such information about the TME can be acquired by measuring the change in the cancer cell volume fraction distribution between two times T0 and T1, e.g., based on MRI images. We demonstrate this principle by using a continuum based multiphase model for tumor progression combined with assimilation of observed data through an ensemble Kalman filter approach which has been extensively and successfully used for updating advanced multiphase flow models in the context of reservoir simulation. Our results based on a synthetic dataset demonstrate how the methodology can be used to extract valuable quantitative information (e.g., interstitial fluid velocity field and fluid pressure, tissue conductivity reflecting ECM status, and effective vasculature conductivity) for which direct measurements may not be possible or impractical.publishedVersio

Mathematical Analysis of Two Competing Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow

Recent experimental work has revealed that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms. One is a chemotactic-driven mechanism where chemokine (chemical component) bounded to the extracellular matrix (ECM) is released and skewed in the flow direction. This leads to higher chemical concentrations downstream which the tumor cells can sense and migrate toward. The other is a mechanism where the flowing fluid imposes a stress on the tumor cells which triggers them to go in the upstream direction. Researchers have suggested that these two migration modes possibly can play a role in metastatic behavior, i.e., the process where tumor cells are able to break loose from the primary tumor and move to nearby lymphatic vessels. In Waldeland and Evje (J Biomech 81:22–35, 2018), a mathematical cell–fluid model was put forward based on a mixture theory formulation. It was demonstrated that the model was able to capture the main characteristics of the two competing migration mechanisms. The objective of the current work is to seek deeper insight into certain qualitative aspects of these competing mechanisms by means of mathematical methods. For that purpose, we propose a simpler version of the cell–fluid model mentioned above but such that the two competing migration mechanisms are retained. An initial cell distribution in a one-dimensional slab is exposed to a constant fluid flow from one end to the other, consistent with the experimental setup. Then, we explore by means of analytical estimates the long-time behavior of the two competing migration mechanisms for two different scenarios: (i) when the initial cell volume fraction is low and (ii) when the initial cell volume fraction is high. In particular, it is demonstrated in a strict mathematical sense that for a sufficiently low initial cell volume fraction, the downstream migration dominates in the sense that the solution converges to a downstream-dominated steady state as time elapses. On the other hand, with a sufficiently high initial cell volume fraction, the upstream migration mechanism is the stronger in the sense that the solution converges to an upstream-dominated steady state.publishedVersio

Identification of the flux function of nonlinear conservation laws with variable parameters

Machine learning methods have in various ways emerged as a useful tool for modeling the dynamics of physical systems in the context of partial differential equations (PDEs). Nonlinear conservation laws (NCLs) of the form ut+f(u)x = 0 play a vital role within the family of PDEs. A main challenge with NCLs is that solutions contain discontinuities. That is, one or several jumps of the form (uL(t), uR(t)) with uL ≠ uR may move in space and time such that information about f(u) in the interval associated with this jump is not present in the observation data. Moreover, the lack of regularity in the solution u (x,t) prevents use of physics informed neural network (PINN) and similar methods. The purpose of this work is to propose a method to identify the nonlinear flux function f (u, β) with variable parameters of an unknown scalar conservation law (*)ut + f(u, β)x = 0 with u as the dependent variable and β as the parameter. In a recent work we introduced a framework coined ConsLaw-Net that combines a symbolic multi-layer neural network and an entropy-satisfying discrete scheme to learn the nonlinear, unknown flux function f(u; β) for various fixed β. Learning the flux function with variable parameters, marked as f(u, β), is more challenging since it requires an understanding of the relationship between the variable u and parameter β as well. In this work we demonstrate how to couple ConsLaw-Net to the Linear Regression Neural Network (LRNN) to learn the functional form of the two variable function f(u, β). In addition, ConsLaw-Net is here further developed and made more generic by using a refined discrete scheme combined with a more general symbolic neural network (S-Net). We experimentally demonstrate that the upgraded ConsLaw-Net integrated with LRNN is well-suited for learning tasks, achieving more accurate identification than existing learning approaches when applied to problems that involve general classes of flux functions f(u, β). The investigations of this work are restricted to the class of polynomial, rational functions.publishedVersio

On the numerical discretization of a tumor progression model driven by competing migration mechanisms

In this work we explore a recently proposed biphasic cell-fluid chemotaxis-Stokes model which is able to represent two competing cancer cell migration mechanisms reported from experimental studies. Both mechanisms depend on the fluid flow but in a completely different way. One mechanism depends on chemical signaling and leads to migration in the downstream direction. The other depends on mechnical signaling and triggers cancer cells to go upstream. The primary objective of this paper is to explore an alternative numerical discretization of this model by borrowing ideas from [Qiao et al. (2020), M3AS 30]. Numerical investigations give insight into which parameters that are critical for the ability to generate aggressive cancer cell behavior in terms of detachment of cancer cells from the primary tumor and creation of isolated groups of cancer cells close to the lymphatic vessels. The secondary objective is to propose a reduced model by exploiting the fact that the fluid velocity field is largely dictated by the draining fluid from the leaky tumor vasculature and collecting peritumoral lymphatics and is more weakly coupled to the cell phase. This suggests that the fluid flow equations to a certain extent might be decoupled from the cell phase equations. The resulting model, which represents a counterpart of the much studied chemotaxis-Stokes model model proposed by [Tuval, et al. (2005), PNAS 102], is explored by numerical experiments in a one-dimensional tumor setting. We find that the model largely coincides with the original as assessed through numerical solutions computed by discrete schemes. This model might be more amenable for further explorations and analysis. We also investigate how to exploit the weaker coupling between cell phase dynamics and fluid dynamics to do more efficient calculations with fewer updates of the fluid pressure and velocity field.publishedVersio

An Integrated Modeling Approach for Vertical Gas Migration Along Leaking Wells Using a Compressible Two-Fluid Flow Model

Gas migration behind casings can occur in wells where the annular cement barrier fails to provide adequate zonal isolation. A direct consequence of gas migration is annular pressure build-up at wellhead, referred to as sustained casing pressure (SCP). Current mathematical models for analyzing SCP normally assume gas migration along the cemented interval to be single-phase steady-state Darcy flow in the absence of gravity and use a drift-flux model for two-phase transport through the mud column above the cement. By design, such models do not account for the possible simultaneous flow of gas and liquid along the annulus cement or the impact of liquid saturation within the cemented intervals on the surface pressure build-up. We introduce a general compressible two-fluid model which is solved over the entire well using a newly developed numerical scheme. The model is first validated against field observations and used for a parametric study. Next, detailed studies are performed, and the results demonstrate that the surface pressure build-up depends on the location of cement intervals with low permeability, and the significance of two-phase co-current or counter-current flow of liquid and gas occurs along cement barriers that have an initial liquid saturation. As the magnitude of the frictional pressure gradient associated with counter-current of liquid and gas can be comparable to the relevant hydrostatic pressure gradient, two-phase flow effects can significantly impact the interpretation of the wellhead pressure build-up

Identification of nonlinear conservation laws for multiphase flow based on Bayesian inversion

Conservation laws of the generic form ct+f(c)x=0 play a central role in the mathematical description of various engineering related processes. Identification of an unknown flux function f(c) from observation data in space and time is challenging due to the fact that the solution c(x, t) develops discontinuities in finite time. We explore a Bayesian type of method based on representing the unknown flux f(c) as a Gaussian random process (parameter vector) combined with an iterative ensemble Kalman filter (EnKF) approach to learn the unknown, nonlinear flux function. As a testing ground, we consider displacement of two fluids in a vertical domain where the nonlinear dynamics is a result of a competition between gravity and viscous forces. This process is described by a multidimensional Navier–Stokes model. Subject to appropriate scaling and simplification constraints, a 1D nonlinear scalar conservation law ct+f(c)x=0 can be derived with an explicit expression for f(c) for the volume fraction c(x, t). We consider small (noisy) observation data sets in terms of time series extracted at a few fixed positions in space. The proposed identification method is explored for a range of different displacement conditions ranging from pure concave to highly non-convex f(c). No a priori information about the sought flux function is given except a sound choice of smoothness for the a priori flux ensemble. It is demonstrated that the method possesses a strong ability to identify the unknown flux function. The role played by the choice of initial data c0(x) as well various types of observation data is highlighted.publishedVersio

Solving Nonlinear Conservation Laws of Partial Differential Equations Using Graph Neural Networks

Nonlinear Conservation Laws of Partial Differential Equations (PDEs) are widely used in different domains. Solving these types of equations is a significant and challenging task. Graph Neural Networks (GNNs) have recently been established as fast and accurate alternatives for principled solvers when applied to standard equations with regular solutions. There have been few investigations on GNNs implemented for complex PDEs with nonlinear conservation laws. Herein, we explore GNNs to solve the following problem ut + f(u, β)x = 0 where f(u, β) is the nonlinear flux function of the scalar conservation law, u is the main variable, and β is the physical parameter. The main challenge of nonlinear conservation laws is that solutions typically create shocks. That is, one or several jumps in the form (uL, uR) with uL ≠ uR moving in space and probably changing over time such that information about f(u) in the interval associated with this jump is not present in the observation data. We demonstrate that GNNs could achieve accurate estimates of PDEs solutions based on new initial conditions and physical parameters within a specific parameter range.publishedVersio

Fluid-sensitive migration mechanisms predict association between metastasis and high interstitial fluid pressure in pancreatic cancer

A remarkable feature in pancreatic cancer is the propensity to metastasize early, even for small, early stage cancers. We use a computer-based pancreatic model to simulate tumor progression behavior where fluid-sensitive migration mechanisms are accounted for as a plausible driver for metastasis. The model has been trained to comply with in vitro results to determine input parameters that characterize the migration mechanisms. To mimic previously studied preclinical xenografts we run the computer model informed with an ensemble of stochastic-generated realizations of unknown parameters related to tumor microenvironment only constrained such that pathological realistic values for interstitial fluid pressure (IFP) are obtained. The in silico model suggests the occurrence of a steady production of small clusters of cancer cells that detach from the primary tumor and form isolated islands and thereby creates a natural prerequisite for a strong invasion into the lymph nodes and venous system. The model predicts that this behavior is associated with high interstitial fluid pressure (IFP), consistent with published experimental findings. The continuum-based model is the first to explain published results for preclinical models which have reported associations between high IFP and high metastatic propensity and thereby serves to shed light on possible mechanisms behind the clinical aggressiveness of pancreatic cancer.publishedVersio