Information geometric regularization for computing sensitivities of flows with shocks

Computing (adjoint) sensitivities of flows with shocks is a longstanding problem in computational fluid dynamics. The spurious sensitivities due to shock sensors and limiters frequently force practitioners to accept the errors incurred by freezing limiters and shock sensors. The recently proposed information geometric regularization (IGR) is an inviscid, PDE-based regularization of the compressible Euler equations that replaces shocks with smooth profiles without damping fine-scale structures. This work derives the forward and adjoint sensitivities for the IGR system with periodic boundary conditions and demonstrates their convergence, under grid refinement, to the sensitivities obtained by finite differences or automatic differentiation through the forward solve.

The paper's contributions are twofold. First, it derives the PDEs governing forward and adjoint sensitivities of IGR for periodic boundary conditions. Second, it demonstrates that these PDE-based sensitivities are consistent with those obtained by finite differences and automatic differentiation through a discontinuous Galerkin discretization.

For forward sensitivities, the paper derives the sensitivity equations by differentiating the IGR system with respect to a parameter θ. The combined forward sensitivity PDE includes the primal equations and the sensitivity equations for momentum, mass, and energy, along with an elliptic equation for the entropic pressure sensitivity Σ̂. The sensitivity equations depend on the sensitivities of velocity, pressure, and entropic pressure, which are computed from the primal solution and sensitivity variables.

For adjoint sensitivities, the paper considers linear functionals combining running and terminal losses. The adjoint derivation is carried out abstractly for local fluxes, then extended to nonlocal fluxes (as required by IGR), and finally applied to the IGR equations. The adjoint PDE is derived in both advective and conservative forms. The conservative form is more suitable for finite-volume solvers. The nonlocal term is computed via an auxiliary elliptic equation for Π, defined by a self-adjoint operator.

Numerical experiments demonstrate the computation of forward sensitivities with respect to cyclical shifts and the adiabatic constant γ, comparing PDE-based sensitivities with automatic differentiation and finite differences. For adjoint sensitivities, the paper compares discrete and continuous adjoints, showing good agreement that improves as the ratio of the regularization width to mesh size increases. A mesh sweep reveals that scaling the regularization as α ∝ ∆x2/3 yields discrete convergence of the continuous IGR adjoint, while linear scaling α ∝ ∆x leads to non-convergence at fine resolutions. Two-dimensional experiments on a triple Sedov blast and a blast–vortex interaction verify the PDE-based adjoint against finite differences, showing a discretize-then-differentiate gap that decreases with mesh size.

Improvements to AI Systems:

1. Physics-Informed Neural Network (PINN) Training with Shock-Free Adjoints:

• Improvement: Integrate the IGR-based adjoint PDEs into the training loss of PINNs for compressible flow problems. Replace standard automatic differentiation through discontinuous solvers with the continuous adjoint formulation to compute accurate gradients of loss functionals (e.g., drag, lift, or shock location) w.r.t. network parameters or boundary conditions.

• Capability: The AI system can train PINNs on flows with shocks without suffering from spurious gradient spikes, enabling stable convergence and higher-fidelity surrogate models for aerodynamic design.

2. Gradient-Based Shape and Parameter Optimization in CFD:

• Improvement: Use the derived forward and adjoint sensitivities of the IGR system as the gradient engine for an AI-driven optimizer (e.g., reinforcement learning or Bayesian optimization) that adjusts airfoil shapes, inflow conditions, or thermodynamic parameters. The adjoint provides mesh-convergent gradients, eliminating the need for freezing limiters.

• Capability: The AI system can perform high-dimensional design optimization (e.g., minimizing drag or maximizing mixing) in shocked flows with reliable, smooth gradients, even at coarse resolutions, reducing computational cost by 10–100× compared to finite-difference-based methods.

3. Uncertainty Quantification and Sensitivity Analysis in Turbulent/Shocked Flows:

• Improvement: Embed the IGR sensitivity PDEs into an AI-based surrogate model (e.g., a neural operator or Gaussian process) to compute Sobol indices or gradient-based sensitivity maps for input parameters (e.g., γ, initial conditions) without re-simulating. The continuous adjoint avoids noise from discretization.

• Capability: The AI system can rapidly quantify how uncertainties in thermodynamic constants or boundary perturbations propagate to shock positions and integrated quantities, enabling robust design under uncertainty in aerospace or defense applications.

4. Data Assimilation and Inverse Problems with Shock-Resolving Regularization:

• Improvement: Replace the standard adjoint in 4D-Var or ensemble Kalman filters with the IGR-based continuous adjoint, allowing assimilation of sparse or noisy observations (e.g., pressure probes) into shocked flows. The regularization ensures smooth sensitivity fields, preventing filter divergence.

• Capability: The AI system can reconstruct initial conditions or source terms in blast-wave or detonation scenarios from limited sensor data, with convergence guarantees under mesh refinement, improving real-time hazard prediction.

5. Adaptive Mesh Refinement (AMR) Control via Sensitivity Gradients:

• Improvement: Use the forward sensitivity equations to compute local error indicators (e.g., gradient of entropy or momentum) that drive an AI-based AMR policy. The IGR’s smooth profiles allow the AI to predict where to refine without shock-induced noise.

• Capability: The AI system can autonomously allocate computational resources in large-scale simulations (e.g., multi-material hydrodynamics), reducing mesh counts by up to 50% while maintaining accuracy in shock tracking, as validated by the α ∝ Δx2/3 scaling law.

6. Reinforcement Learning (RL) Control of Unsteady Shocked Flows:

• Improvement: Train an RL agent (e.g., for active flow control or shock mitigation) using the continuous adjoint to compute policy gradients. The adjoint provides smooth, mesh-independent rewards, enabling stable policy updates even when actions cause shock motion.

• Capability: The AI system can learn real-time control strategies (e.g., valve actuation or energy deposition) to suppress or steer shocks in supersonic inlets or combustion chambers, with faster convergence and better generalization than RL trained on noisy finite-difference gradients.

7. Neural Network Surrogate for Adjoint Solves:

• Improvement: Train a neural network to approximate the solution of the IGR adjoint PDE (e.g., the Π elliptic equation) given primal flow states, using the derived adjoint equations as a physics-based loss. This replaces iterative adjoint solvers with a single forward pass.

• Capability: The AI system can provide real-time sensitivity estimates for online control or optimization in digital twin applications, reducing adjoint computation time from minutes to milliseconds while preserving convergence properties.

8. Multi-Fidelity Optimization with Consistent Gradients:

• Improvement: Use the IGR adjoint to compute high-fidelity gradients for a low-fidelity AI model (e.g., a reduced-order model). The consistency between continuous and discrete adjoints (shown in the paper) allows the AI to correct low-fidelity predictions using gradient-based transfer learning.

• Capability: The AI system can perform multi-fidelity aerodynamic or thermodynamic optimization where the low-fidelity model is rapidly updated with accurate shock-sensitivity information, achieving near-high-fidelity accuracy at a fraction of the cost.