Source-linked AI summary
DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators
Zhiping Mao, Lu Lu, Olaf Marxen, Tamer A. Zaki, George E. Karniadakis
TL;DR
Hypersonic flow with finite-rate chemistry couples shock-driven temperature changes, species dissociation, and flow properties in a computationally demanding multiphysics problem. The paper uses pre-trained DeepONets within DeepM&Mnet architectures to assimilate sparse measurements and infer the coupled fields. The framework supports accurate data assimilation and can produce solutions in a fraction of a second, while convergence is not guaranteed for nonlinear inverse problems and the model assumes ideal-gas species behavior.
Problem
Hypersonic chemically reacting flow couples multiple physical scales and species dynamics, while conventional simulation requires solving computationally demanding coupled equations.
Method
DeepM&Mnet combines pre-trained DeepONets with sparse sensor data, operator-loss physics incorporation, and parallel or series architectures to infer coupled flow and chemistry fields.
Results
The framework accurately predicts the coupled non-equilibrium chemistry and flow from limited input data and can produce solutions in a fraction of a second.
Takeaways & Limitations
Pre-training encodes prior solutions so DeepM&Mnet can integrate models and scarce sensor data for multiphysics data assimilation.
Takeaways & Limitations
Convergence is not guaranteed for the nonlinear inverse problem, regularization may be needed, and the gas model assumes species behave as ideal gases.
Abstract
from arXiv · showhide
In high-speed flow past a normal shock, the fluid temperature rises rapidly triggering downstream chemical dissociation reactions. The chemical changes lead to appreciable changes in fluid properties, and these coupled multiphysics and the resulting multiscale dynamics are challenging to resolve numerically. Using conventional computational fluid dynamics (CFD) requires excessive computing cost. Here, we propose a totally new efficient approach, assuming that some sparse measurements of the state variables are available that can be seamlessly integrated in the simulation algorithm. We employ a special neural network for approximating nonlinear operators, the DeepONet, which is used to predict separately each individual field, given inputs from the rest of the fields of the coupled multiphysics system. We demonstrate the effectiveness of DeepONet by predicting five species in the non-equilibrium chemistry downstream of a normal shock at high Mach numbers as well as the velocity and temperature fields. We show that upon training, DeepONets can be over five orders of magnitude faster than the CFD solver employed to generate the training data and yield good accuracy for unseen Mach numbers within the range of training. Outside this range, DeepONet can still predict accurately and fast if a few sparse measurements are available. We then propose a composite supervised neural network, DeepM&Mnet, that uses multiple pre-trained DeepONets as building blocks and scattered measurements to infer the set of all seven fields in the entire domain of interest. Two DeepM&Mnet architectures are tested, and we demonstrate the accuracy and capacity for efficient data assimilation. DeepM&Mnet is simple and general: it can be employed to construct complex multiphysics and multiscale models and assimilate sparse measurements using pre-trained DeepONets in a "plug-and-play" mode.
1. Introduction
High-speed chemically reacting flows couple shock-driven fluid dynamics with finite-rate chemistry across multiple scales, making conventional simulation and sparse-data assimilation difficult. The paper introduces DeepONets and DeepM&Mnet to approximate nonlinear operators and combine pre-trained models with limited measurements.
- Motivation: Shock heating can trigger chemical dissociation, while composition changes affect energy balance, viscosity, and heat conduction.These interactions create coupled flow and chemistry downstream of a normal shock.
- Motivation: Finite-rate chemistry requires species transport equations with reaction source terms, which can introduce numerical stiffness and complicate high-Mach simulations.Methods designed for low-Mach combustion are not directly applicable to hypersonics.
- Approach: DeepONets approximate nonlinear operators offline, enabling fast online predictions for new input functions and serving as building blocks for multiphysics models.The proposed DeepM&Mnet framework combines these operators with data for seven coupled fields downstream of a normal shock.
- Results: 100,000X speedup is reported for trained DeepONets compared with the CFD solver for flow and five-species non-equilibrium chemistry at Mach numbers between 8 and 10.The five species’ densities span 8 orders of magnitude downstream of the shock.
- Results: Out-of-range Mach-number predictions are initially unsatisfactory but become very good when a few measurements are combined with pre-trained DeepONets through a supervised neural network.The framework also includes parallel and series DeepM&Mnet architectures for assimilating measurements across different subsets of variables.
2. Problem setup and data generation
The study models a two-dimensional, five-species chemically reacting gas downstream of a normal shock using nondimensional compressible-flow equations and generates datasets across Mach numbers 8–10. The simulations resolve coupled species, velocity, temperature, transport, and energy effects on a fine domain, including an eight-order-of-magnitude range for ρNO.
- Fluid-mechanical model: The governing model advances compressible Navier–Stokes conservation equations for mass, momentum, energy, and five reacting species in two spatial dimensions.Finite-rate reaction source terms describe species production and consumption.
- Thermodynamic and transport closure: Finite-rate reaction source terms and composition-dependent viscosity and thermal conductivity couple chemical composition to the flow and energy equations.Reaction rates are supplied by MUTATION, while transport properties are also computed with that library.
- Fluid-mechanical model: The five-species air mixture comprises N, O, N2, NO, and O2, with pressure and temperature related through an ideal-gas mixture equation of state.Species partial pressures sum to the mixture pressure.
- Thermodynamic and transport closure: Temperature is computed from internal energy and species densities, accounting for translational, rotational, vibrational, and molecular formation-energy contributions.The iterative calculation is performed within the MUTATION library.
- Numerical setup: The computational domain extends 0.02 downstream of the shock with 160 streamwise and 21 transverse grid points, using high-order compact differences and third-order Runge–Kutta time integration.The inflow uses a Dirichlet condition and the transverse boundaries are periodic.
- Data generation: The dataset contains 400 trajectories for M∞∈[8, 10], split into 240 training and 60 testing trajectories, while ρNO spans 8 orders of magnitude across a thin boundary layer.The simulations use post-shock conditions as inflow conditions and retain the prescribed composition across the short streamwise domain.
3. Developing DeepONets as building blocks
The paper develops independently trained DeepONets that map between the five chemical-species densities and the velocity-temperature fields, forming building blocks for DeepM&Mnets. These models accurately predict coupled fields, support fast extrapolation with sparse output data, and enable subsequent data assimilation.
- DeepONet architecture: DeepONets are developed as building blocks that predict remaining coupled fields from either flow variables or five chemical-species densities.The two mappings are GU,T: species densities → [U, T] and GρN2,O2,N,O,NO: [U, T] → five species densities.
- DeepONet architecture: Each unstacked DeepONet uses a branch net to encode sampled input functions and a trunk net to encode output locations.The branch net observes the input function at fixed sensors, while the trunk net encodes locations for the output functions.
- Training procedure: Logarithms of the density data are used during training, whereas velocity and temperature use their original data.The logarithmic transformation is identified as an important consideration for achieving the reported density accuracy.
- Prediction within the training range: Predictions of velocity, temperature, and five species densities agree closely with independent reference data, including densities spanning orders of magnitude.Training and testing errors are commensurate for velocity and temperature, while species-density predictions accurately capture large variations downstream of the shock.
- Extrapolation and data assimilation: Outside the training Mach-number range, initial predictions can shift from reference solutions, but a supervised add-on network improves them using only a few output measurements.The add-on network is trained against sparse sensor data and is designed to require no exhaustive retraining; satisfactory predictions are reported with three data points.
- Extrapolation and data assimilation: The trained DeepONets provide the operators used to couple flow and chemistry in DeepM&Mnets for efficient data assimilation.The paper presents the DeepONets as prerequisites for combining pretrained operators with sparse measurements when the full input function is unavailable.
4. DeepM&Mnet framework: architectures and results
DeepM&Mnet couples pre-trained DeepONets with sparse measurements to infer the seven coupled flow and chemistry fields. Parallel and series architectures use operator losses and optional conservation or regularization terms to support accurate data assimilation.
- Framework: DeepM&Mnet couples pre-trained DeepONets with sensor data to predict the seven coupled flow and chemical-species fields.The framework samples Mach numbers in [8, 10] and uses the pre-trained operators as building blocks.
- Parallel architecture: The parallel architecture takes spatial coordinates as input and predicts velocity, temperature, and five species densities, while DeepONets provide reciprocal operator outputs.Its loss combines discrepancies between neural-network outputs, DeepONet outputs, and available measurements.
- Parallel architecture: The parallel loss can include L2 regularization and a global mass-conservation constraint in addition to data and operator losses.The conservation term represents the steady-state global mass condition.
- Parallel results: The parallel network produces very accurate predictions, while pre-trained DeepONet outputs remain smooth and accurate without regularization.The authors interpret the pre-trained operators as an effective physics-encoding regularizer in these tests.
- Parallel results: With L2 regularization, enforcing global conservation significantly improves accuracy for all variables, whereas without regularization its accuracy benefit is not appreciable.The reported comparison uses mean square errors for the parallel architecture.
- Series architecture: The series architecture requires only sparse velocity and temperature data, then passes inferred fields through chained DeepONets to recover all seven variables.Its outputs are naturally regularized because the inputs to the flow operators come from upstream species operators.
- Series results: For the series architecture, adding global conservation to L2 regularization yields much more accurate predictions than L2 regularization alone.The reported series experiment uses 5 data points for both velocity and temperature.
5. Conclusion
The paper develops DeepM&Mnet for hypersonic flow with coupled finite-rate chemistry and sparse data assimilation. It combines pre-trained DeepONets into parallel and series architectures, including a flow-only-data configuration and optional global conservation.
- Problem: Hypersonic flow behind a normal shock couples high-temperature dissociation chemistry with flow changes and steep species-density variations.The five chemical-species densities span eight orders of magnitude downstream of the shock.
- Framework: DeepM&Mnet combines pre-trained DeepONets with a supervised neural network to assimilate sparse measurements and infer coupled fields.The framework is tested for both in-range and out-of-range Mach-number conditions.
- Architectures: The framework provides parallel and series architectures, with the series variant requiring only sparse velocity and temperature data.Global mass conservation can be added as an accuracy-improving and stabilizing constraint.
- Implications: DeepM&Mnet targets data-poor multiphysics regimes and can produce solutions in a fraction of a second after offline training.The authors present it as a potential paradigm for complex multiphysics and multiscale computational problems.
Appendix A. DeepONet predictions for the chemical species
Appendix A constructs separate DeepONets for three chemical species using selected molecular-species densities as inputs. The models predict new species-density conditions and report training and testing losses.
- Architecture: Three independently trained DeepONets predict ρN from ρN2, ρO from ρO2, and ρNO from both ρN2 and ρO2.The appendix treats each target output with a separate operator model.
- Interpretation: The operator inputs omit velocity, temperature, and other species because their influence is encoded in each trained operator.This is an operator-level representation of the coupled physics rather than a claim that those variables are physically irrelevant.
- Training setup: The chemical-density data used for training are logarithms of the original densities.This transformation is applied to all density data in the appendix configuration.
- Training setup: The DeepONets use 4 × 100 hidden layers in both branch and trunk networks, adaptive ReLU activation, learning rate 8 × 10^-4, and 70000 epochs.These are the stated training parameters for the appendix models.
- Results: Figure A.16 reports predictions for ρN, ρO, and ρNO together with their training and test losses.The appendix uses these results to demonstrate prediction capacity for new conditions.
Appendix B. Another type of series DeepM&Mnet architecture
Appendix B tests a series DeepM&Mnet that starts from sparse data for the five species densities and predicts flow variables through chained pre-trained operators. Regularization is necessary for stable training in this configuration.
- Architecture: The appendix interchanges the roles of density and flow in the series architecture to examine regularization during data assimilation.The network uses density measurements as its available inputs.
- Architecture: A trained neural network maps spatial coordinates to the five species densities, which are passed to the flow DeepONets to predict velocity and temperature.The loss combines measurement errors with discrepancies between network outputs and DeepONet outputs.
- Experiment: The experiment uses 6 data points for the densities of all species with the same training parameters as subsection 4.2.The reported evaluation compares mean square errors under regularization and global conservation settings.
- Results: Without regularization, training remains unstable even with global conservation; strong regularization or weak regularization plus conservation stabilizes it.The tested regularization weights are ωR = 10^-4 and ωR = 10^-5, respectively.