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Physics-informed neural networks for inverse problems in supersonic flows
Ameya D. Jagtap, Zhiping Mao, Nikolaus Adams, George Em Karniadakis
TL;DR
The paper tackles ill-posed inverse problems for two-dimensional supersonic flows using sparse density-gradient and boundary data. It employs PINNs and XPINNs constrained by the Euler equations, entropy conditions, and positivity requirements, and reports accurate reconstructions across expansion and shock-wave cases, with adaptive formulations outperforming vanilla PINNs in discontinuous settings.
Problem
Inverse supersonic compressible-flow problems are ill-posed and difficult for traditional solvers, despite their importance in specialized aerospace-vehicle design.
Method
PINNs and domain-decomposed XPINNs infer density, velocity, and pressure from Schlieren density gradients, partial boundary data, Euler equations, entropy conditions, and positivity constraints.
Results
Adaptive-activation PINNs and XPINNs match exact density, velocity, and pressure solutions for discontinuous oblique-shock cases, whereas vanilla PINNs produce large errors.
Takeaways & Limitations
The study demonstrates PINNs and XPINNs for inverse two-dimensional expansion, oblique-shock, and bow-shock problems using limited experimental and boundary information.
Takeaways & Limitations
The study assumes γ = 1.4 for the air-flow expansion-wave example and notes that Euler equations may admit multiple entropy pairs.
Abstract
from arXiv · showhide
Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we have data available for density gradients from Schlieren photography as well as data at the inflow and part of wall boundaries. These inverse problems are notoriously difficult and traditional methods may not be adequate to solve such ill-posed inverse problems. To this end, we employ the physics-informed neural networks (PINNs) and its extended version, extended PINNs (XPINNs), where domain decomposition allows deploying locally powerful neural networks in each subdomain, which can provide additional expressivity in subdomains, where a complex solution is expected. Apart from the governing compressible Euler equations, we also enforce the entropy conditions in order to obtain viscosity solutions. Moreover, we enforce positivity conditions on density and pressure. We consider inverse problems involving two-dimensional expansion waves, two-dimensional oblique and bow shock waves. We compare solutions obtained by PINNs and XPINNs and invoke some theoretical results that can be used to decide on the generalization errors of the two methods.
1. Introduction
The paper addresses ill-posed inverse shock-wave problems in two-dimensional supersonic compressible flows using PINNs and XPINNs with partial experimental and boundary data. It extends PINN methods to expansion, oblique-shock, and bow-shock problems while incorporating entropy and positivity conditions.
- Motivation: PINNs are investigated as mesh-free methods for solving ill-posed inverse problems with noisy, sparse, or multi-fidelity data.The introduction identifies PINNs as an alternative to traditional numerical methods for computational science and engineering problems.
- Motivation: Inverse supersonic-flow problems are difficult because compressible Euler equations admit shocks, while traditional solvers are expensive and often require difficult-to-specify boundary conditions.Far-field or symmetry conditions can enlarge computational domains, and numerical boundary conditions may cause instability.
- Approach: The study extends PINN ideas from one-dimensional cases to two-dimensional shock-wave problems and considers arbitrarily shaped physical domains.This addresses limitations associated with large specified computational domains in traditional methods.
- Approach: The inverse problems use density-gradient information, limited primitive-variable data, and global constraints to infer density, velocity, and pressure fields.The framework uses PINNs and XPINNs to reconstruct states for oblique shocks, expansion waves, and bow shocks.
- Study scope: The paper compares PINNs and XPINNs across expansion-wave, oblique-shock, and bow-shock configurations.Figure 1 schematically distinguishes the oblique-shock, expansion-wave, and bow-shock settings considered.
2. Governing Equations
The governing model is the two-dimensional compressible Euler system closed by a polytropic-gas equation of state. Entropy conditions select physically relevant weak solutions, while positivity constraints keep density and pressure positive during training.
- Euler equations: The compressible Euler equations are written in conservative form as ∂tU + ∇·G(U) = 0 over a two-dimensional spatial domain.The conservative variables and fluxes are represented through U and G(U).
- Euler equations: The state variables include density, pressure, two velocity components, and total energy, with an equation of state used to close the system.The paper uses a polytropic-gas equation of state.
- Steady formulation: The steady-state Euler equations are used for the inverse supersonic-flow problems, with γ representing the ratio of specific heats and γ = 1.4 for air.The neural-network training uses the nondimensional form of the equations.
- Entropy conditions: Because the Euler equations admit multiple weak solutions, entropy conditions are imposed to select physically relevant viscosity solutions.Viscosity solutions are described as zero-viscosity limits and satisfy entropy conditions.
- Entropy conditions: At discontinuities, admissible nonsmooth solutions satisfy the entropy condition in a distributional sense, although the Euler equations may have multiple entropy pairs.The paper notes that the displayed entropy-entropy flux pair is a common choice for Euler equations.
- Physical constraints: Training enforces positivity of density and pressure by using max(α, ρ) and max(α, p), where 0 < α << 1.This is intended to avoid negative density and pressure values.
3. Methodology
The methodology uses feed-forward PINNs and XPINNs to infer supersonic-flow fields from partial boundary and Schlieren density-gradient data while enforcing governing physics and physical admissibility.
- Physics-informed neural networks: PINNs learn density, velocity, and pressure by combining data-mismatch terms with residual terms for Euler conservation laws and entropy conditions.The loss incorporates primitive-variable data, Schlieren density-gradient data, and PDE residuals constructed using automatic differentiation.
- Physics-informed neural networks: Density and pressure positivity are enforced during training by replacing each network output with its maximum against a small positive constant α.This avoids handling negative density and pressure values during optimization.
- Extended physics-informed neural networks: XPINNs decompose the computational domain into arbitrary non-overlapping subdomains, assigning local networks and residual points according to solution complexity.Subdomain-wise networks can increase representation capacity and reduce approximation, optimization, or generalization errors when the decomposition and point placement are selected appropriately.
- Extended physics-informed neural networks: XPINN losses retain the PINN structure within each subdomain and add interface terms that stitch neighboring subdomain solutions together.The interface terms include averaged primitive-variable mismatches and residual continuity conditions.
- Optimization methods: Training uses Adam first and switches to L-BFGS after the loss becomes small, exploiting Adam’s robustness and L-BFGS’s efficiency for smooth solutions.L-BFGS is described as a quasi-Newton method with second-order accuracy, whereas Adam is first-order but generally more robust.
4. Computational results
The computational study applies PINNs and XPINNs to inverse expansion-wave and shock-wave problems using partial interior and boundary data. Across the examples, adaptive activations, dynamic weights, and domain decomposition improve predictions and theoretical generalization behavior, particularly for discontinuous solutions.
- Study design: The study compares PINN and XPINN methods across expansion-wave, oblique-shock, and bow-shock inverse problems.The comparison focuses on ill-posed supersonic-flow problems with interior and boundary data rather than computational cost.
- 4.1. Expansion wave problem: The expansion-wave inverse problem uses Euler and entropy residuals, inlet conditions, density-gradient data in a small region, and sparse wall-pressure data.The weighted loss combines these terms, with dynamic weights optionally applied to the mean-squared-error components.
- 4.1. Expansion wave problem: Adaptive activations or dynamic weights improve PINN accuracy for pressure and velocity, while density predictions remain in good agreement with the exact solution across configurations.Dynamic weights are reported as the most accurate among the PINN results for the expansion-wave case.
- 4.1. Expansion wave problem: XPINN divides the expansion-wave domain into two subdomains with 300 interface points and produces results slightly better than PINN.The XPINN uses the same network architecture in each subdomain and adaptive activation functions.
- 4.1. Expansion wave problem: XPINN has a better theoretical generalization bound than PINN for the expansion-wave problem, while complexity is largest for PINN in the more complex first subdomain.XPINN complexity is slightly larger in the second subdomain because of the reduced training data set.
- 4.2. Oblique shock problem: For the oblique-shock problem, vanilla PINN gives large errors, whereas adaptive activations or dynamic weights yield primitive-variable predictions that match the exact solution well.XPINN is also compared with PINN, and the oblique shock is captured with and without the entropy condition.
- 4.3. Detached bow shock problem: XPINN shows better predictive accuracy than PINN for the detached bow-shock problem, while dynamic weights for PINN converge at different scales across loss terms.The reported dynamic weights ω4 and ω5 converge around 10^3, whereas ω2 and ω3 converge around 10^1 and 10^2.
5. Summary
The paper solves inverse two-dimensional supersonic flow problems with PINNs and XPINNs using sparse experimental and boundary data while enforcing physical constraints. It compares predictive accuracy and theoretical generalization behavior across expansion and shock-wave cases.
- PINNs and XPINNs solve inverse expansion and compression-wave problems in two-dimensional supersonic flows.The considered cases include expansion waves, oblique shock waves, and bow shock waves.
- The models use density-gradient data from Schlieren imaging, inflow data, and pressure data on the wall boundary.
- Entropy conditions and positivity constraints on density and pressure are enforced alongside the compressible Euler equations.The entropy conditions target viscosity solutions, while positivity avoids nonphysical density and pressure values.
- The study compares PINN and XPINN predictive accuracy and uses theoretical generalization bounds to assess model complexity and generalization.The theoretical comparisons particularly address the expansion-wave and oblique-shock test cases.
Appendix A. Pedagogical example: Inverse problem for the two-dimensional Euler equations with smooth solutions
A smooth two-dimensional Euler example reconstructs density, velocity, and pressure from density-gradient data along y = x, one pressure trace, and inflow conditions. The resulting PINN predictions are reported as good for all states.
- The example uses the two-dimensional Euler equations with periodic boundary and initial conditions.
- The initial state has density 1.0 + 0.2 sin(π(x + y)) and constant velocity components u = 0.7, v = 0.3 with pressure p = 1.0.
- The exact solution transports the sinusoidal density profile while retaining u = 0.7, v = 0.3, and p = 1.0.
- The inverse setup uses density-gradient information only along y = x, one pressure value p(−1, −1, t), and inflow conditions.
- The PINN produces good predictions for density, velocity, and pressure in the smooth-solution example.