Source-linked AI summary
Thermodynamically consistent physics-informed neural networks for hyperbolic systems
Ravi G. Patel, Indu Manickam, Nathaniel A. Trask, Mitchell A. Wood, Myoungkyu Lee, Ignacio Tomas, Eric C. Cyr
TL;DR
Existing PINNs face discretization-related difficulties involving boundary conditions, conservation, and regularity when applied to hyperbolic systems and shock-physics inverse problems. The paper adapts a least-squares space-time control-volume scheme, adds entropy and thermodynamic biases, and applies the framework to noisy equation-of-state inference. The resulting framework improves solution quality and supports equation-of-state modeling from realistic gas and metal simulation data, while optimization error remains a major convergence barrier.
Problem
PINNs face challenges with residual weighting, solution regularity, conservation, and forward or inverse modeling of shock-physics systems requiring accurate equations of state.
Method
The paper adapts a space-time control-volume discretization for PINNs and adds finite-volume, entropy, total-variation, and thermodynamic regularizers for conservation-law and equation-of-state problems.
Results
The framework obtains improved accuracy and solution quality for conservation laws and solves equation-of-state inverse problems using realistic noisy data from Argon DSMC and copper molecular-dynamics simulations.
Takeaways & Limitations
Thermodynamic inductive biases allow neural-network equation-of-state models to address practical engineering inverse problems involving rarefied gases and non-fluid materials.
Takeaways & Limitations
Optimization error and the absence of established convergence with respect to neural-network size remain major barriers to matching traditional finite-element and finite-volume methods.
Abstract
from arXiv · showhide
Physics-informed neural network architectures have emerged as a powerful tool for developing flexible PDE solvers which easily assimilate data, but face challenges related to the PDE discretization underpinning them. By instead adapting a least squares space-time control volume scheme, we circumvent issues particularly related to imposition of boundary conditions and conservation while reducing solution regularity requirements. Additionally, connections to classical finite volume methods allows application of biases toward entropy solutions and total variation diminishing properties. For inverse problems, we may impose further thermodynamic biases, allowing us to fit shock hydrodynamics models to molecular simulation of rarefied gases and metals. The resulting data-driven equations of state may be incorporated into traditional shock hydrodynamics codes.
1. Introduction
The paper develops a control-volume extension of PINNs for hyperbolic conservation laws, targeting discretization challenges in forward and inverse shock-physics problems. It combines finite-volume ideas with entropy constraints and applies thermodynamic biases to equation-of-state inference from simulation data.
- Training challenges can prevent DNNs from producing convergent forward PDE schemes, despite their ability to represent low-regularity data such as shocks and contact discontinuities.
- Shock-hydrodynamics models require accurate equations of state because materials traverse broad phase-space regions that include phase transitions.
- Weighted least-squares collocation in PINNs requires residual weighting, continuous solutions, and additional penalties to address conservation and hyperbolic forward problems.
- The proposed space-time least-squares control-volume discretization reduces hyperparameters, improves solution quality, and supports entropy inequalities through connections to finite-volume methods.
- The work claims the first thermodynamically consistent treatment of inverse shock-physics problems that assimilates traditional finite-volume methodology into PINN-like methods.
- The framework studies canonical hyperbolic systems and extracts equations of state from noisy DSMC Argon and molecular-dynamics copper-shock data.
2. Space-time Integral Form PDE Formulation
The formulation treats hyperbolic conservation laws and entropy conditions in space-time integral form. Extended space-time fluxes enable generalized divergence expressions and an integral balance law over arbitrary control volumes.
- The paper considers conservation laws for a conserved vector quantity u on a space-time domain, with flux F and an upwind-associated boundary subset Γ−.
- Weak interpretations can admit nonphysical multivalued solutions, motivating viscosity solutions defined through the zero-viscosity limit.
- A convex mathematical entropy η and entropy flux q form an entropy-flux pair when they satisfy the stated compatibility identity.
- The entropy inequality holds for viscosity solutions, with equality only in smooth regions, distinguishing mathematical entropy from specific entropy.
- Appending u to the flux and η to the entropy flux produces extended space-time fluxes whose generalized divergences express the conservation and entropy laws.
- Applying the Gauss divergence theorem converts the generalized divergence formulation into an integral conservation law over compact space-time control volumes.
3. Physics-informed neural networks
The section reframes PINNs for hyperbolic conservation laws through a space-time control-volume formulation, addressing boundary-condition, conservation, regularity, and shock-related issues. Entropy, total-variation, and artificial-viscosity penalties provide additional mechanisms for physically admissible solutions.
- 3.1. Classical PINNs: Point collocation least squares: Classical PINNs use point-collocation least squares, requiring calibrated penalty weights for PDE residuals, initial conditions, and boundary conditions.The solution is represented by a neural network optimized with first-order methods.
- 3.1. Classical PINNs: Point collocation least squares: Point-collocation formulations require continuous solution spaces, creating difficulties for shocks and other reduced-regularity problems.They also require penalty weighting to obtain coercivity, but neural networks lack a fixed mesh providing the necessary geometric information.
- 3.2. Control Volume PINNs: cvPINNs generate residuals from the space-time integral conservation law over disjoint control volumes, while conditional flux evaluation incorporates boundary and initial conditions directly.This removes the initial- and boundary-condition penalty parameters.
- 3.2. Control Volume PINNs: The control-volume formulation imposes conservation naturally, eliminating separate conservation penalties, but its neural-network facet traces require approximate quadrature.Composite trapezoidal or midpoint quadrature is refined so quadrature error remains below optimization error.
- 3.2. Control Volume PINNs: L2 residual minimization is not generally known to recover the physically valid viscosity solution, so the method pragmatically retains L2 optimization and adds entropy and total-variation penalties.For systems, the appropriate residual norm remains largely open; L1 recovery results cited in the text apply to scalar conservation laws and Hamilton–Jacobi equations.
- 3.2. Control Volume PINNs: Artificial viscosity is a comparison mechanism because, at sufficiently small cell widths, optimization error may prevent the loss from representing its effect.The authors therefore motivate alternative vanishing-viscosity mechanisms.
- 3.2. Control Volume PINNs: The method supplements the cvPINN loss with artificial-viscosity, entropy-inequality, and total-variation-diminishing penalties to address shocks, contacts, and nonphysical oscillations.The total-variation bound is desirable for one-dimensional scalar conservation laws, although it is not guaranteed for multidimensional scalars or general hyperbolic systems.
4. Least squares control volume scheme for inverse problems
The inverse framework parameterizes partially unknown conservation-law fluxes, including equations of state, and jointly fits state variables and flux parameters from scattered observations. Thermodynamic entropy constraints distinguish increasingly black-box EOS models and help maintain physically admissible solutions.
- Inverse problem formulation: The inverse problem represents an unknown flux as Fσ, where σ may encode material properties, candidate models, or a neural network.The framework covers unknown viscosity, multiscale closures, and equations of state.
- Inverse problem formulation: State-network parameters and flux parameters are minimized simultaneously using scattered, potentially partial observations of the state.The observations are defined at a point set D within the domain.
- Equation-of-state estimation: For shock hydrodynamics, the EOS may be either a known model form with an estimated parameter or a black-box neural network for extreme-energy materials.The motivating application is the estimation of complete equations of state in one-dimensional Euler flow.
- Equation-of-state estimation: A complete-EOS parameterizes specific entropy, from which temperature and pressure are computed through Gibbs relations, closing the conservation law.The complete-EOS description provides well-defined entropy pairs through its entropy parameterization.
- Thermodynamic constraints: Entropy inequality constraints are required for physically admissible solutions because violating them can destroy Euler hyperbolicity and the minimum principle of specific entropy.The constraints are imposed as part of the thermodynamic parameterization and loss design.
- Thermodynamic constraints: The study compares perfect-gas, entropy-constrained neural-network, and unconstrained neural-network EOS parameterizations as increasingly black-box models.The sequence is used to characterize the role of physical inductive biases in generalizability and model stability.
5. Results: Forward solution of hyperbolic problems
The forward tests show that cvPINNs needs entropy and TVD penalties to select physical shock solutions and suppress oscillations in nonlinear hyperbolic problems. The method also works on unstructured meshes and is less sensitive to boundary and initial-condition weighting than traditional PINNs.
- Entropy condition: Entropy penalization rejects the nonphysical rarefaction shock in Burgers’ problem and recovers the entropy solution for both ϵent = 0.01 and 1.0.With ϵent = 0, training remains at the initialized rarefaction shock, which violates the entropy inequality.
- TVD condition: Increasing ϵTVD from 0 to 1 incrementally reduces oscillations in the Sod shock solution, yielding an essentially non-oscillatory result at ϵTVD = 1.The profiles use density, velocity, and pressure at t = 1 and compare against the analytical solution.
- Artificial Viscosity: For Buckley-Leverett, entropy+TVD regularization recovers the correct shock speed but retains a slight overshoot, while artificial viscosity gives a more dissipative improvement.For Euler, entropy+TVD eliminates shock oscillations and is less dissipative than viscous penalization.
- Unstructured mesh: cvPINNs produces similar solutions on structured and unstructured triangular space-time meshes for a Sod shock tube.The unstructured calculation uses identical parameters and is compared with a comparable 100×100 Cartesian grid.
- PINNs comparison: Traditional PINNs are highly sensitive to initial- and boundary-condition penalty weights, whereas cvPINNs identifies the Burgers shock front, with TVD regularization needed to avoid shock oscillations.The comparison evaluates solutions with minimal loss over 10 trials.
6. Inverse problems: equations of state for shock physics
The paper infers equations of state from DSMC and molecular-dynamics data, then tests them in traditional finite-difference solvers. Model-form assumptions and thermodynamic regularization determine the balance among generalizability, stability, and data requirements.
- Method: The study extracts EOSs with cvPINNs and evaluates them by deploying the learned models in a traditional finite-difference Euler solver.The solver assessment tests accuracy after deployment rather than only fit quality during training.
- Parameterized EOS: A parameterized perfect-gas EOS recovers low error from one DSMC shock-tube sample and generalizes well to unseen initial conditions.The resulting finite-difference simulations agree well with DSMC data regardless of the number of training simulations.
- Neural-network EOS: An unregularized neural-network EOS fits one DSMC case but can lose hyperbolicity, making the finite-difference solution unstable on test states traversing elliptic regions.Additional training expands the hyperbolic region and improves agreement with DSMC data.
- Regularized EOS: Thermodynamically regularized neural-network EOSs balance generalizability with assumptions on the EOS form and provide stable discretized PDEs.The regularized parameterization is described as stable even in small-data limits.
- Copper shock hydrodynamics: For copper impact data, enlarging the training set to encompass more (ρ, e) states improves the learned-EOS test solution and avoids the split-shock behavior seen with distant training states.The molecular-dynamics training data cost roughly 600 cpu·hours and produced an EOS applicable across a wide range of shock conditions.
7. Conclusion
The paper presents a physics-informed machine-learning framework built around a space-time control-volume scheme and thermodynamic regularizers. It applies the framework to noisy inverse problems while noting that neural-network optimization error remains a major obstacle to convergence competitive with traditional methods.
- Conclusion: The framework incorporates a space-time control-volume scheme to improve accuracy and solution quality for conservation laws solved with neural networks.The paper also introduces regularizers for thermodynamic inductive biases in inverse problems.
- Conclusion: Thermodynamic biases are used to infer equations of state from realistic, noisy data sets corresponding to practical engineering problems.The paper identifies extensions to more complex closures as a future application.
- Limitations: Optimization error and the lack of demonstrated convergence with increasing neural-network size remain major challenges to competing with traditional finite-element and finite-volume methods.The paper presents qualitatively correct and physically meaningful solutions but does not resolve this convergence barrier.
Appendix A. Hyper-parameters
The appendix lists neural-network architectures and optimization settings used for the reported results, along with parameter tables for the benchmark problems.
- Neural-network settings: PDE-solution networks use dense architectures with width 64 and depth 8.EOS networks use separate architectures described in the same appendix.
- Neural-network settings: EOS networks use dense architectures with width 4 and depth 4, tanh activations, and the Adam optimizer.Adam is used for all minimization problems.
- Benchmark parameters: Tables A.1–A.4 list parameters for the Burgers rarefaction, Sod shock, Buckley–Leverett and Euler Riemann, and Burgers shock problems.The tables correspond to Figures 1–3 and 5.
Appendix B. Data sets
The appendix identifies the simulation parameters used for the DSMC Sod shock problem and the LAMMPS copper-impact problem.
- Data sets: The data-set appendix covers both DSMC simulations of the Sod shock problem and LAMMPS simulations of copper impact.These simulations supply the data used in the paper’s inverse-problem studies.
Appendix C. Finite difference scheme
The Euler equations with learned equations of state are solved using a viscously regularized centered-difference scheme on a fine Cartesian mesh. The selected scheme and parameters produce stable, sharp solutions for the considered problems.
- Euler equations with learned equations of state are solved using a viscously regularized centered-difference scheme on a fine Cartesian mesh.The grid uses points {x_i} with spacing Δx and advances at times {t_n} with timestep Δt.
- The scheme and chosen parameters produce stable and sharp solutions for the problems considered.