Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations

This paper investigates the sensitivity of Physics-Informed Neural Networks (PINNs) to the choice between conservative and non-conservative PDE formulations when solving problems involving shocks and discontinuities, using benchmark cases like the Burgers and Euler equations to evaluate their effectiveness in bridging the gap between these two approaches.

The Gist

The Big Problem: Two Ways to Tell a Story

Imagine you are trying to describe how a crowd of people moves through a hallway. You can tell this story in two different ways:

1. The "Box" Method (Conservative Form): You count how many people are in specific boxes (sections of the hallway) and track how many people enter or leave each box. If a person leaves Box A, they must enter Box B. This method is great at tracking "stuff" (like mass or energy) even when things get chaotic, like a sudden stampede or a wall of people crashing into each other (a shock).
2. The "Flow" Method (Non-Conservative Form): You describe the speed and direction of the people at every single point. You say, "The person here is moving fast," and "The person there is moving slow." This is often easier to write down and understand intuitively, but if a sudden crash happens, this method gets confused. It might say people disappeared or appeared out of nowhere because it doesn't strictly track the "boxes."

The Catch: In the world of high-speed physics (like supersonic jets or explosions), things often crash into each other, creating "shocks."

• If you use the Box Method, you get the right answer every time, but the math is heavy and slow.
• If you use the Flow Method, the math is lighter, but when a crash happens, the answer is usually wrong. It's like trying to predict a car crash by just looking at the speedometers; you miss the impact.

The Old Solution: Adding "Glue"

For decades, scientists tried to fix the "Flow Method" by adding Artificial Viscosity. Think of this as adding a thick, sticky glue to the air. When a crash happens, the glue smears the crash out over a few inches so the math doesn't break.

• The Problem: It works, but it makes the picture blurry. The sharp edge of the shock becomes a fuzzy blob. Also, you have to guess exactly how much glue to add. Too little, and the math explodes; too much, and the result is useless.

The New Hero: PINNs with "Adaptive Viscosity"

This paper introduces a new superhero: Physics-Informed Neural Networks (PINNs) equipped with Adaptive Weight and Viscosity (AWV).

Think of a PINN not as a calculator, but as a super-smart student who is trying to learn the laws of physics by reading a textbook (the equations) and looking at a few snapshots of reality (data).

Here is how this new student is different:

1. They Learn the Rules: Instead of just memorizing numbers, the student is forced to satisfy the laws of physics (the equations) while they learn.
2. The Magic Trick (Adaptive Viscosity): This is the paper's big breakthrough.
   • In the old days, scientists had to manually decide how much "glue" (viscosity) to add to smooth out a crash.
   • This new student learns how much glue to add on the fly.
   • If the math gets shaky near a crash, the student automatically adds just enough glue to stabilize it. If the flow is smooth, they remove the glue to keep the picture sharp.
   • It's like a self-driving car that automatically adjusts its suspension: stiff on smooth roads for speed, soft on bumpy roads for comfort, without the driver touching a knob.

The Experiment: Putting It to the Test

The authors tested this "super-student" on three classic physics problems:

1. The Smooth Wave (Burgers Equation): When the flow is smooth (no crashes), the student solved it perfectly, whether they used the "Box" method or the "Flow" method. Result: No surprise here.
2. The Sudden Crash (Burgers with a Shock): When they introduced a sudden crash:
   • The old "Flow" method (without glue) failed completely.
   • The old "Flow" method (with manual glue) worked but was blurry.
   • The PINN Student used the "Flow" method but figured out the perfect amount of glue automatically. It solved the crash accurately, just as well as the heavy "Box" method.
3. The Supersonic Jet (Euler Equations): They tested this on a wedge-shaped object moving faster than sound.
   • Again, the PINN student handled the "Flow" method beautifully, capturing the sharp shock waves without the blurriness of the old methods.

The Takeaway

The "Box" method has always been the gold standard for crashes because it's safe, but it's hard to use for complex problems (like multi-phase flows or rotating systems) where you can't easily write the "Box" equations.

This paper proves that PINNs with Adaptive Viscosity can use the easier "Flow" equations and still get the "Box" level of accuracy.

In simple terms:
Imagine you are trying to paint a picture of a storm.

• Old Way: You have to use a heavy, rigid frame (Conservative) to keep the rain from splashing everywhere. It's accurate but hard to carry.
• New Way: You use a flexible frame (Non-Conservative), but you have a magical brush (PINN-AWV) that knows exactly when to hold the paint steady and when to let it flow. You get the same perfect picture, but you can paint much more complex scenes that the rigid frame couldn't handle.

Conclusion: This technology bridges the gap. It allows scientists to use the simpler, more intuitive math for complex problems while still getting the precise, shock-capturing results that were previously impossible without the heavy, conservative math.

Technical Summary

1. Problem Statement

In Computational Fluid Dynamics (CFD), the formulation of Partial Differential Equations (PDEs) as either conservative or non-conservative forms significantly impacts numerical accuracy, particularly in flows involving shocks and discontinuities.

• Conservative Form: Expresses laws (mass, momentum, energy) in divergence form ($\partial_t U + \nabla \cdot F(U) = 0$). It is the standard for capturing shocks correctly because it inherently satisfies the Rankine-Hugoniot (RH) jump conditions, ensuring correct shock speeds and conservation of physical quantities.
• Non-Conservative Form: Derived via the chain rule (e.g., using primitive variables like velocity and pressure). While mathematically equivalent to the conservative form in smooth regions, it fails to preserve conservation laws across discontinuities. This leads to:
  • Incorrect shock speeds.
  • Smeared or non-existent shocks.
  • Violation of entropy conditions.
• The Gap: Many complex physical phenomena (e.g., multi-phase flows, shallow water with variable topography, non-Newtonian fluids) are inherently non-conservative. Traditional numerical solvers struggle with these forms in the presence of shocks, often requiring complex, problem-specific stabilization techniques (e.g., artificial viscosity tuning, path-conservative methods).
• Hypothesis: The authors investigate whether Physics-Informed Neural Networks (PINNs), specifically those equipped with adaptive mechanisms, can solve both conservative and non-conservative formulations with equal accuracy, effectively "bridging the divide" that plagues traditional numerical methods.

2. Methodology

The study employs a specific variant of PINNs called PINNs with Adaptive Weight and Viscosity (PINNs-AWV).

A. Core Architecture (PINNs-AWV)

Standard PINNs minimize a loss function comprising data mismatch and PDE residuals. However, near shocks, PDE residuals become extremely large, destabilizing training. The PINNs-AWV architecture addresses this via two adaptive mechanisms:

1. Adaptive Weighting: The weights of the loss terms are dynamically adjusted based on gradient norms. This reduces the influence of the PDE loss in regions with high residuals (shocks), preventing the optimizer from being dominated by these singularities.
   • Formula: $\lambda_{PDE} = \frac{1}{\|\nabla_\theta L_i\| + \epsilon}$
2. Adaptive Viscosity: Instead of using a fixed artificial viscosity (which smears shocks), the network learns an optimal, spatially varying viscosity coefficient ($\nu$) via a sub-network.
   • A "viscous loss" term ($L_\nu = \nu^2$) is added to the total loss to minimize viscosity where it is not needed, preserving sharp shock resolution.
   • The modified PDE residual includes the learned viscosity: $R = \partial_t u + u \partial_x u - \nu \partial_{xx} u$.

B. Benchmark Problems

The authors tested the methodology on three distinct problems:

1. Burgers' Equation:
   • Smooth IC: To verify baseline performance.
   • Discontinuous IC (Riemann Problem): To test shock capturing capabilities without artificial viscosity tuning.
2. Sod Shock Tube (Unsteady 1D Euler Equations):
   • A classic test for compressible flow involving a shock wave, contact discontinuity, and expansion fan.
   • Compared Conservative vs. Non-Conservative forms using both PINNs-AWV and traditional numerical methods (HLLC solver, MUSCL scheme).
3. Supersonic Flow over a Wedge (Steady 2D Euler Equations):
   • Simulated a Mach 2 flow over a 10-degree wedge to generate an oblique shock.
   • Tested the ability to resolve oblique shocks in a steady-state, multi-dimensional setting.

3. Key Contributions

1. Formulation Independence: The primary contribution is demonstrating that PINNs-AWV can solve the non-conservative form of hyperbolic PDEs with the same accuracy as the conservative form. Traditional numerical methods fail to capture the correct shock speed in the non-conservative form without extensive, problem-specific tuning.
2. Unified Framework: The study proposes a unified framework where the choice between conservative and non-conservative forms does not dictate the solver's success. This is crucial for complex systems (like multi-phase flows) where a conservative form may not exist or is difficult to derive.
3. Automatic Stabilization: Unlike traditional methods that require manual tuning of artificial viscosity coefficients (often suboptimal), PINNs-AWV learns the optimal viscosity distribution required to stabilize the solution while minimizing shock smearing.
4. Validation of RH Conditions: The results show that PINNs-AWV implicitly satisfies the Rankine-Hugoniot conditions even when the governing equations are written in non-conservative form, a feat rarely achieved by standard non-conservative numerical schemes.

4. Results

• Burgers' Equation (Smooth): Both conservative and non-conservative forms yielded identical results across all methods (PINNs and numerical), confirming that the distinction is irrelevant for smooth flows.
• Burgers' Equation (Discontinuous):
  • Traditional Non-Conservative: Failed to propagate the shock (shock speed = 0).
  • Traditional Non-Conservative + Fixed Viscosity: Captured the shock but with significant smearing (diffusion).
  • PINNs-AWV (Non-Conservative): Successfully captured the shock at the correct speed and location with minimal smearing, matching the conservative PINN solution.
• Sod Shock Tube:
  • Traditional non-conservative numerical schemes were highly diffusive compared to conservative schemes.
  • PINNs-AWV produced accurate solutions for both forms. The pressure contours and shock locations were nearly identical for conservative and non-conservative PINN formulations, whereas traditional non-conservative solvers showed significant deviation.
• Supersonic Wedge:
  • PINNs-AWV successfully resolved the oblique shock for both formulations.
  • The shock was resolved within approximately 3 grid points, consistent with high-fidelity expectations.
  • The non-conservative PINN solution matched the conservative solution and the exact analytical solution, whereas traditional non-conservative solvers were deemed too unstable to include without extensive tuning.

5. Significance and Conclusion

This work challenges the conventional wisdom in CFD that non-conservative forms are inherently unsuitable for shock-capturing problems.

• Theoretical Implication: It suggests that the failure of non-conservative schemes is not a fundamental mathematical limitation but a limitation of traditional discretization methods (finite difference/volume) that cannot adaptively handle the discontinuities in the derivative terms.
• Practical Impact: By enabling the use of non-conservative forms (which are often more intuitive or the only available form for complex physics like multi-phase flows or magnetohydrodynamics) without sacrificing shock accuracy, PINNs-AWV opens new avenues for simulating complex physical phenomena.
• Future Outlook: The authors conclude that while traditional solvers require complex, case-specific stabilization (limiters, artificial viscosity tuning), the PINNs-AWV approach offers a robust, unified, and data-free (in terms of training data, relying only on physics) alternative for solving nonlinear PDEs with discontinuities.

In summary: The paper demonstrates that by integrating adaptive weighting and learned viscosity into the PINN framework, the "divide" between conservative and non-conservative formulations can be bridged, allowing for accurate shock resolution regardless of the mathematical form of the governing equations.