Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks

1. Introduction

• High-dimensional parametric surrogate models can become computationally intractable as their complexity grows exponentially.Such models are used for fast exploration when inverse problems require multiple flow-solver calls.Source: §1 · p. 2
• Moving flow fronts challenge tensor representations, which may require many basis vectors and lose their computational advantage.The paper notes that this difficulty also affects problems with localized solution features.Source: §1 · p. 3
• Neural networks may offer a more flexible representation for moving discontinuities in transport-equation solutions.This is presented in comparison with tensor formats.Source: §1 · p. 3
• The paper aims to establish a neural-network framework for multiphase flow in porous media, including a basic framework for two-phase flow.Source: §1 · pp. 3–4
• PINNs approximate PDE solutions with neural networks and minimize PDE and boundary-condition residuals over collocation and training points.The introduction notes that collocation-point choice is crucial for accuracy.Source: §1 · pp. 3–4
• The contributions include residual-based adaptation for flow-front prediction and training-point adaptation for initial and boundary conditions and separate PDE collocation in coupled systems.Source: §1 · p. 4

2.1. Model problem for two-phase flow in porous media

• Darcy’s law describes flow velocity using permeability, viscosity, and the pressure gradient, while incompressible fluids obey mass conservation.The formulation also assigns boundary conditions for velocity, pressure, and fraction, plus initial conditions for the fraction.Source: §2.1 · p. 5
• The model distinguishes the two fluids with a fraction function c, and its evolution follows an advection equation.The fraction takes values 1 and 0 for the respective fluids; its time derivative is denoted c_t.Source: §2.1 · p. 5
• Pressure Dirichlet conditions are prescribed at the inlet and outlet, impermeable boundaries have zero normal velocity, and inlet flow has a fraction boundary condition.Source: §2.1 · pp. 5–6

2.2. PINN

• A fully connected neural network approximates the solution, mapping temporal or spatial inputs to the output through learned layer weights, biases, and nonlinear activations.The formulation obtains PDE residuals by automatic differentiation and minimizes them alongside boundary-condition residuals at finite collocation and training points.Source: §2.2 · p. 6

2.3. PINN structure and loss function definition for the model problem

• PINNs use three neural networks for velocity, pressure, and fraction, with spatial and temporal inputs and automatic-differentiation residuals for the governing equations.The velocity network has one, two, or three outputs according to problem dimension.Source: §2.3 · pp. 6–7
• The weighted loss combines velocity, fraction, and pressure boundary-condition terms with residual terms for the three differential equations.The fields are obtained by minimizing this loss with respect to the network parameters.Source: §2.3 · pp. 7–8

3. Collocation points adaptivity

• Classical mesh adaptation includes h-refinement by adding nodes, r-adaptation by relocating nodes, and p-adaptation by increasing element polynomial degree.Adaptation drivers include error, PDE residual, and solution features; feature-based refinement can over-refine some features while ignoring others.Source: §3 · pp. 8–9
• PINN point adaptation is comparatively cheap because the method is meshless: point locations can change without updating mesh connectivity or handling element volumes.Derivative approximation is independent of collocation-point position, leaving the method to identify where additional points are needed.Source: §3 · pp. 9–10
• RAR adds points with the largest residuals, but can cluster them in narrow regions and over-refine moving sharp fronts while ignoring other solution features.Figure 2 illustrates points concentrated in a small region.Source: §3 · p. 10, Figure 2 · p. 11
• A residual-based probability density controls point spread: more points are drawn where residuals are high and fewer where they are low.A tolerance filters small residuals, and normalization over the space-time domain is calculated by Monte Carlo integration.Source: §3 · pp. 11–13
• For coupled differential equations, each PDE residual receives its own collocation points, while data points are enriched similarly to capture initial and boundary conditions.Source: §3 · p. 12
• The algorithm evaluates residuals on dense sets, builds probability functions, draws and adds points, optimizes for a fixed number of iterations, and checks residual means against tolerances.Source: §3 · p. 13

4.1. One dimensional injection

• The one-dimensional experiment uses 2,500 points for fixed PINN and starts RAR and new adaptivity at 1,600 points; all three take near 200 seconds to converge.The adaptive and RAR cases add points every 50 BFGS iterations until reaching 2,500 points.Source: §4.1 · p. 16
• The new adaptive PINN gives the best one-dimensional front results among the three cases, while RAR provides no significant improvement.The authors suggest RAR’s concentration of added points in a small space-time region may make optimization harder and increase generalization error.Source: Figure 5 · p. 17, §4.1 · p. 17
• The new adaptive case produces pressure profiles closer to the analytical solution than fixed PINN or RAR, whose prediction is far off near t = 0.The profiles are compared at t = 0, 0.15, 0.25, and 0.5.Source: Figure 6 · p. 18, §4.1 · p. 18
• For the one-dimensional fraction equation, added collocation points cluster near the front where residuals are higher; points for the other PDEs remain almost random across the domain.The authors attribute the broader distribution for the other PDEs to residuals spread across the domain without sharp solution features.Source: §4.1 · p. 20
• In both examples, new adaptivity reduces the discrepancy between training and testing loss, indicating reduced generalization error.For the one-dimensional case, the fixed-point loss deviation is marginally decreased by RAR and significantly decreased by the new method.Source: §4.1 · p. 20, §4.2 · p. 26
• In the one-dimensional adaptive case, the fraction-function initial and boundary-condition losses are highest, consistent with their discontinuous values at t = 0.The loss terms otherwise show broadly similar convergence trends.Source: §4.1 · p. 21

4.2. Two dimensional central injection

• The two-dimensional test uses a unit-area square with a central elliptic inlet and four outlets; four five-layer networks approximate pressure, fraction, and two velocity components.Each network has 20 neurons per hidden layer.Source: §4.2 · pp. 21–22
• In the two-dimensional central-injection example, the new adaptive algorithm predicts the flow-front position better than the other cases.The front positions are compared in both x and y directions against the analytical solution.Source: Figure 12 · p. 24, Figure 13 · p. 25, §4.2 · p. 25

• The method draws new collocation points from a PDE-residual-based probability density and enriches points for each PDE and for initial and boundary conditions.Source: §5 · p. 26
• In 1D and 2D two-phase-flow problems, the adaptive technique produced better results than PINN with fixed collocation points and improved over RAR.Source: §5 · p. 26
• The adaptive technique can be viewed as regularization of the neural networks, reducing generalization error.Source: §5 · pp. 26–27