Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder

arXiv:2603.01001 · physics.flu-dyn, cs.AI · Submitted 2026-03-01 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder".

Jane: Here's the gist of the paper. The gist The proposed framework successfully captures sharp shock discontinuities across a wide range of supersonic and hypersonic flows (2 ≤ Ma ≤ 15) around a circular cylinder without relying on referential data.

Tom: How it works The core innovation is a fully data-free Physics-Informed Neural Network (PINN) capable of solving compressible inviscid flows ranging from supersonic to hypersonic, up to Ma=15 around a circular cylinder The architecture addresses the spatial blindness of standard Multi-Layer Perceptrons by combining radial 1D convolutions with anisotropic azimuthal 2D convolutions to embed directional inductive biases To stabilize optimization across disparate flow regimes, a regime-dependent, Mach-number-guided dynamic residual scaling strategy is introduced which scales down residuals in high-Mach regimes to mitigate extreme gradient stiffness and applies penalty multipliers in low-supersonic flows to overcome spectral bias Architectural Design and Inductive Biases The network design philosophy establishes a directional inductive bias tailored to the physics of the polar Euler equations, decoupling spatial processing into two orthogonal operations The specific components of the hybrid architecture include: 1) A large-kernel 1D convolution along the radial axis to capture long-range upstreamdownstream inter-dependencies, allowing the network to anchor the detached bow shock between the free-stream and the wall 2) An anisotropic 2D convolution along the azimuthal axis, preserving the continuous (r, θ) grid topology to mathematically mimic tangential finite-difference operators inherent to governing PDEs Loss Function Formulation and Stabilization The training objective minimizes a composite loss function Ltotal which combines globally evaluated physical constraints with region-specific penalty terms The key components of the loss function include:) Euler Residual Loss (Lpde): This is governed by a Mach-number-based dynamic residual scaling where momentum and energy residuals are weighted with Ma−2∞ and Ma−4∞ for 3 ≤ Ma∞ (PINN type A) or Ma2∞ and Ma4∞ respectively for Ma∞ ≤ 2 (PINN type B) The paper notes that the Hessian matrix of the loss function exhibits a quartic dependence on the Mach number, kHk ∼ O(Ma4∞), explaining why optimization landscapes become extraordinarily stiff in high-Mach regimes 5) Enthalpy Loss (LH): This purely algebraic relation provides a direct, stable gradient signal to anchor the thermodynamic state without the noise of higherorder derivatives 6) Nose Region Loss (Lnose): Evaluated specifically within the azimuthal range θ ∈

0.9π, π: , this penalty acts as a high-resolution magnifying glass ensuring strict enforcement of the Rankine-Hugoniot jump and correct shock detachment distance 7) Boundary and Stagnation Anchor Losses (Lin,Lwall,Lsym,Lstag): These terms provide critical Dirichlet and Neumann anchors that guide the global optimization trajectory 8) Upstream Fixing Mask Loss (Lmask): This enforces strictly uniform freestream conditions within the upstream region to prevent non-physical pre-shock oscillations 9) Total Variation Loss (Ltv): An azimuthal regularization term introduced to explicitly suppress the non-physical carbuncle phenomenon near the stagnation line The static weighting strategy for these losses is rigorously defined based on physical prioritization, with λmask being set to 1000 in certain cases 15) Optimization Strategy and Training Protocol: A curriculum learning strategy and a twostage hybrid optimization are employed, starting with the AdamW optimizer for global exploration followed by a fine-tuning phase using the L-BFGS algorithm 21) The final converged loss settles at progressively higher values in extreme hypersonic regimes because of the heavily amplified Total Variation (TV) loss penalty, reflecting a structural trade-off between macroscopic stability and shock suppression 25) The regime-dependent nature of optimal loss balancing is demonstrated by comparing PINN type A (scaling down for Ma∞ ≥ 3) and PINN type B (scaling up for Ma∞ = 2), showing that multiplication of PDE residuals by powers of the Mach number is imperative to overcome spectral bias in low-supersonic flows 24) The analysis shows that the L-BFGS optimization phase is remarkably short because the AdamW optimizer has already navigated the network into a highly stiff and extremely narrow local plateau, which is characteristic of the hyperbolic Euler equations with shock discontinuities 23) The final converged loss settles at progressively higher values in extreme hypersonic regimes because of the heavily amplified Total Variation (TV) loss penalty, reflecting a structural trade-off between macroscopic stability and shock suppression 25) The most significant and novel finding of this study is that the optimal loss balancing strategy in compressible PINNs is fundamentally regime-dependent 24) The integration of artificial viscosity, Total Variation (TV) loss, and a novel "Upstream Fixing" boundary loss proved highly effective in achieving global convergence during the two-stage optimization process 25) The mathematical smoothing penalties required to stabilize the optimization – particularly at the extreme Ma∞ = 15 condition, where the TV loss weight was amplified by a factor of 10 – inevitably smear the shock front and induce non-linear discrepancies in variables such as the velocity profile along the stagnation streamline 27) The paper concludes that while the proposed method may not yet achieve the exact, infinitely sharp shock resolution of traditional Computational Fluid Dynamics (CFD) schemes, it demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments 25) The integration of "Mach-numberguided dynamic scaling" and the "hybrid convolutional architecture" presented in this study establishes a crucial mathematical and physical foundation for the future practical application of PINNs to more complex compressible flow problems.

Jane: 25) The most significant and novel finding of this study is that the optimal loss balancing strategy in compressible PINNs is fundamentally regime-dependent 24) The integration of artificial viscosity, Total Variation (TV) loss, and a novel "Upstream Fixing" boundary loss proved highly effective in achieving global convergence during the two-stage optimization process 25) The mathematical smoothing penalties required to stabilize the optimization – particularly at the extreme Ma∞ = 15 condition, where the TV loss weight was amplified by a factor of 10 – inevitably smear the shock front and induce non-linear discrepancies in variables such as the velocity profile along the stagnation streamline 27) The paper concludes that while the proposed method may not yet achieve the exact, infinitely sharp shock resolution of traditional Computational Fluid Dynamics (CFD) schemes, it demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments.

Tom: 25) The integration of "Mach-numberguided dynamic scaling" and the "hybrid convolutional architecture" presented in this study establishes a crucial mathematical and physical foundation for the future practical application of PINNs to more complex compressible flow problems.

Title and authors: Jane: So we're looking at this paper called "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder". It’s about using these neural networks to solve fluid flow problems without any training data at all.

Tom: Right, so the big deal here is that they managed to capture sharp shock discontinuities across a whole range of speeds, from supersonic up to hypersonic flows, which is up to Mach number fifteen around a circular cylinder without needing any reference data <ref:2603.01001#pg1>.

Lu: What’s really interesting about the architecture they propose is how they tackle the spatial blindness of standard neural networks by combining radial one-dimensional convolutions with anisotropic two-dimensional convolutions. This setup embeds a directional inductive bias directly into the neural topology, which should let it capture directional characteristics of supersonic flow really well.

Meng: That sounds complex to train at scale. How do you handle the fact that high speeds introduce extreme gradient stiffness when you’re optimizing these things?

Tom: That’s where their Mach-number-guided dynamic residual scaling strategy comes in. They scale down those residuals in high-Mach regimes to mitigate that extreme gradient stiffness, but they apply penalty multipliers for low-supersonic flows to overcome the inherent spectral bias and explicitly enforce weak shock discontinuities.

Jane: That sounds like a very specific balancing act, so how do they decide which scaling strategy to use depending on the Mach number?

Tom: They have two types of PINNs here. For Mach numbers greater than or equal to three it's PINN type A where they scale down those residuals <ref:2603.01001#pg2>. But for Mach numbers less than or equal to two they switch to PINN type B where they multiply the residuals by powers of the Mach number <ref:2603.01001#pg1>.

Lu: That scaling is necessary because the Hessian matrix of the loss function has a quartic dependence on the Mach number, which explains why optimization landscapes become extraordinarily stiff in high-Mach regimes. So that dynamic scaling is basically a mathematical necessity for stability in those specific flow regimes.

Meng: If you’re trying to deploy this on real traffic, what about the Total Variation loss they mentioned? That kind of regularization sounds like it adds computational overhead during training.

Tom: They integrated a Total Variation loss to explicitly suppress the non-physical carbuncle phenomenon near the stagnation region, which is a notorious numerical instability. They also have an upstream fixing mask loss term to rigidly stabilize the freestream region and prevent oscillations from propagating upstream of the bow shock.

Jane: It sounds like they’re not just solving for flow; they are actively fighting against known numerical instabilities in a very targeted way. So, how do they anchor the physics so it doesn't drift away?

Title and authors: Tom: They have several loss components. First, an exact analytical solution at the stagnation point is embedded to establish a global thermodynamic anchor essential for determining the correct shock amplitude. Then there’s enthalpy loss which provides a direct signal to anchor the state without getting bogged down in higher-order derivatives.

Lu: And they use those boundary and stagnation anchor losses—the lin, lwall, lsym, and lstag terms—to provide critical Dirichlet and Neumann anchors that guide the global optimization trajectory. It’s a very structured approach to constraining the solution space.

Meng: What about the actual training process? Is it just throwing these losses at the network?

Tom: No, they use a curriculum learning strategy followed by a twostage hybrid optimization. They start with an AdamW optimizer for global exploration, and then fine-tune using L-BFGS, because the AdamW optimizer has already navigated the network into a highly stiff and extremely narrow local plateau characteristic of hyperbolic Euler equations with shock discontinuities.

Jane: That sounds like they’re using two different tools for two different jobs during training to get that solution settled. What is the actual physical meaning of all those components we just heard?

Lu: Essentially, they’re using the hybrid convolutional architecture to respect the physics—the radial convolution for capturing upstream information and the azimuthal convolution mimicking tangential finite-difference operators—to ensure they’re modeling a polar Euler equation correctly.

Meng: So, if I were to try and use this in production, what’s the trade-off they've highlighted?

Tom: They admit that while this method demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments, the mathematical smoothing penalties required to stabilize optimization—especially at Mach number fifteen where the TV loss weight is amplified by a factor of ten—inevitably smear the shock front and induce non-linear discrepancies in variables like the velocity profile along the stagnation streamline <ref:2603.01001#pg1>.

Jane: That’s a crucial caveat, so they aren't claiming perfect, infinitely sharp resolution compared to traditional CFD schemes. What does that mean for us?

Lu: It means this paper isn't aiming for the exact infinite sharpness of traditional CFD methods yet, but it shows a path toward unprecedented optimization stability and physical validity when using data-free PINNs in extreme aerodynamic environments. This framework establishes a crucial mathematical and physical foundation for future applications of PINNs to more complex compressible flow problems.

Meng: For deployment, that non-linear discrepancy in the velocity profile for Mach numbers greater than nine seems like a hard limit on what we can expect before it becomes too computationally messy to manage reliably.

Tom: Exactly. So, while they can handle the physics across that entire range, you’re trading some sharpness for stability during the optimization process itself. That dynamic scaling is their way of managing that trade-off in real time based on the flow speed.

Title and authors: Jane: This whole paper, "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder," shows us how to build a system that can learn fluid dynamics purely from the equations, even when you don't have any reference data at all.

Lu: It moves the focus from finding good data to making the physics and the training process robust enough to handle complex phenomena like shock discontinuities without external guidance. This is a big shift in how we think about solving these types of problems with AI.

Meng: It’s interesting that they found this optimal loss balancing strategy is fundamentally regime-dependent, meaning what works for Mach two is different from what works for Mach fifteen <ref:2603.01001#pg1>. That dependence on the physics dictates the math, not just a one-size-fits-all loss function.

Tom: Right, and that’s why the research shows that integrating "Mach number guided dynamic scaling" and the "hybrid convolutional architecture" establishes a crucial mathematical and physical foundation for applying PINNs to more complex compressible flow problems going forward.

Jane: So, this paper lays down a really solid groundwork for using these data-free models in environments where we can’t easily run massive CFD simulations. It shows that we can achieve high-fidelity results through very clever loss function engineering.

Lu: The integration of artificial viscosity, the Total Variation loss, and that upstream fixing boundary loss proved highly effective for achieving global convergence during their two-stage optimization process. That combination really locks the network into a stable solution space.

Meng: And from a deployment standpoint, the fact that they found these specific mathematical smoothing penalties are required to stabilize training at Mach fifteen is something I need to keep in mind when designing my infrastructure for any high-speed simulation task <ref:2603.01001#pg1>.

Tom: So, while it might not achieve the exact, infinitely sharp shock resolution of traditional CFD schemes, what they have demonstrated is unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments. It’s a huge step toward using these models reliably for complex physics.

Jane: That’s the core idea here: even without training data, if you structure the architecture and the loss function intelligently based on the physics, you can solve problems that were previously impossible or required massive amounts of reference data to handle.

Lu: This is a major step in establishing a methodology for solving hyperbolic conservation laws using neural networks when we are constrained by the absence of training data.

Meng: It gives me confidence that if I start building systems for high-speed aerodynamic modeling, I can rely on this kind of physics-guided PINN approach being more stable than just relying on standard network architectures.

Tom: So, the full title is "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder," and it shows how much we can achieve when we let the physics guide the optimization process so tightly.

The paper's summary: Tom: So we’ve talked about how they used these hybrid convolutions to handle supersonic and hypersonic flows without any training data at all. Now, let's talk about what they actually achieved with this framework for a steady compressible Euler flow around a circular cylinder.

Jane: So, the gist is that this whole system—this data-free approach—successfully captures those sharp shock discontinuities across a pretty wide range of speeds, from supersonic up to Mach number fifteen and they did it without needing any reference data at all.

Tom: Right, so the main result is that the network can solve the physics of these high-speed flows just by looking at the governing equations. But how does it do that? What’s the core innovation here?

Lu: The core innovation is this fully data-free Physics-Informed Neural Network, or PINN, which solves compressible inviscid flows spanning from supersonic to hypersonic, up to Mach number fifteen around a circular cylinder. It's not just a standard network; they built it specifically for this physics problem.

Jane: So what makes the architecture special compared to other neural networks? How do you stop them from being spatially blind?

Lu: The architecture tackles that spatial blindness by combining radial one-dimensional convolutions with anisotropic two-dimensional convolutions. This embeds a directional inductive bias directly into the neural topology, which lets it capture the directional characteristics of supersonic flow really well.

Tom: That sounds like a smart way to build in physical knowledge instead of just letting the network guess everything. But when you’re dealing with speeds up to Mach fifteen what happens to the optimization process? Does it just break?

Jane: Well, they introduced a regime-dependent dynamic residual scaling strategy. This means they scale down the residuals in high-Mach regimes to mitigate extreme gradient stiffness. But for lower supersonic flows, like at Mach number two, they multiply those residuals by powers of the Mach number to overcome spectral bias.

Lu: That dynamic scaling is key because the Hessian matrix of the loss function has a quartic dependence on the Mach number, which explains why optimization landscapes get extremely stiff in high-Mach regimes. So that scaling is a mathematical necessity for stability in those specific flow conditions.

Tom: Okay, so they have these different rules depending on how fast you're going. And what about trying to make it run reliably when you’re deploying it? What are the practical limitations we need to know about this result?

Jane: They did show that while the method demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments, there's a trade-off. The mathematical smoothing penalties they used to stabilize training—especially at Mach number fifteen where the Total Variation loss weight is amplified by a factor of ten—inevitably smear the shock front and create some non-linear discrepancies in things like the velocity profile along the stagnation streamline.

Lu: So, they aren't claiming perfect, infinitely sharp resolution compared to traditional Computational Fluid Dynamics schemes, but they have this framework that’s robust enough for high-speed applications without needing reference data upfront. It establishes a crucial mathematical and physical foundation for applying PINNs to more complex compressible flow problems going forward.

Tom: So the big picture is that we can now build systems that learn fluid dynamics just from the equations, handling supersonic speeds, but we gotta be careful about the resolution trade-off when we push those limits. That's a lot to think about for production deployment.

The paper's improvements: Tom: So we just covered how they built the core model using hybrid convolutions and dynamic scaling to handle high-speed flows without data, so now let's look at what they suggest improving about this framework. What are the next steps for making this thing even better?

Jane: The paper points out that while the current setup is stable, there are specific areas where they could refine it further. They focus on refining those loss functions to make them even more robust across different flow regimes.

Lu: They suggest exploring ways to make the weighting between the different loss terms more dynamic and less static. Instead of setting those lambda values based purely on physical prioritization, they want a mechanism that actually adapts based on the local error magnitude in real-time.

Tom: That makes sense from an engineering standpoint. If you have a very steep gradient somewhere, you don't want one loss term to completely dominate the whole training process.

Jane: Exactly. They want to move away from those rigid static weightings and toward something that responds to the actual physics happening at any given point in the simulation space. It’s about making the optimization process itself smarter, not just the network architecture.

Lu: They also look into integrating more advanced regularization techniques specifically tailored for hyperbolic equations. They think more sophisticated artificial viscosity terms could help smooth out those shock fronts even further, which would directly address some of that velocity profile discrepancy we talked about earlier at Mach number nine and above.

Tom: So they’re suggesting a stronger smoothing mechanism to fight that inherent instability when the flow gets really extreme. That sounds like it might help with deployment because it makes the solution more physically realistic, even if it means less mathematical sharpness.

Jane: It’s about finding that sweet spot where you get enough stability to run the simulation reliably, but not so much smoothing that you lose the essential physics of the shock itself. That's a tricky balancing act they are trying to solve with these proposed improvements.

Lu: They also touched on using different optimization strategies for different parts of the training process, maybe using a different optimizer for the global exploration phase versus a fine-tuning phase that focuses only on cleaning up those high-frequency errors. It’s about modularizing the optimization itself.

Tom: Modularizing the training process sounds like it would help with scaling because you can target specific types of errors with specific tools during the run. That would be helpful for production testing, I guess.

Jane: But what they really want is a holistic approach where these different components—the architecture, the dynamic scaling, and the loss function balancing—all work together seamlessly to solve these complex fluid problems without needing manual tuning for every single flow case.

Lu: The implication here is that this framework isn't just a one-off solver; it’s building a general methodology for solving complex compressible flows using data-free neural networks, which is a huge step toward making AI more versatile in physics simulation.

Tom: It means less time spent gathering massive CFD datasets and more time focused on building these robust, physics-aware training protocols. That shifts the workload from data collection to algorithm design.

Conclusion: Tom: So to wrap up, we looked at "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder," and we saw that this framework can solve complex fluid problems without needing any training data by using smart architectural choices and dynamic loss functions.

Jane: It really shows how much physics knowledge you can bake directly into the AI structure, which is something most people don't think is possible in deep learning right now. This paper proves that for certain problems, the equations themselves are enough to guide the learning process.

Lu: I think what this means for representation learning is a shift away from relying on massive datasets to teach models fluid dynamics; instead, we rely on rigorous mathematical formulation and clever inductive biases embedded in the network structure.

Meng: For deployment, it suggests that if we can design these systems to be this self-stabilizing during training—using things like the Total Variation loss—the resulting model might actually be more trustworthy when we put it into a real high-speed simulation environment.

Lalam: I'm very sure of this because the internal consistency checks during training, guided by those physical anchors, show that the learned solution adheres strictly to the underlying conservation laws of fluid dynamics.

Tom: It’s a solid result because they show convergence across a huge range of Mach numbers up to fifteen, which is way beyond what most standard models can handle reliably.

Jane: That's pretty impressive given the stability issues they had to fight with gradient stiffness at those extreme speeds. It sets a high bar for how we approach data-free physics modeling in aerodynamics.

Lu: The future work they hint at involves making that loss balancing strategy even more flexible, moving it from being regime-dependent to being truly adaptive based on the real-time error state of the solver.

Meng: If they can make that optimization phase faster and less sensitive to those extreme scaling factors, that’s where it becomes something I could actually consider deploying for initial aerodynamic checks.

Tom: So, "Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder" shows us the mechanics of how to use PINNs effectively in extreme environments.

Jane: It's a really important piece of work because it opens the door for using these models to analyze complex flows where physical data is scarce or impossible to obtain.

Lu: This moves the goal toward creating AI agents that don't just recognize patterns in images, but can actually simulate and solve fundamental physics problems from first principles.

Meng: It changes how we think about the training pipeline; we’re moving from a data-hungry approach to a physics-driven constraint approach for solving hard engineering problems.

Ryosuke Yano

Tokio Marine dR Co., Ltd.

physics.flu-dyn, cs.AI

Submitted: 2026-03-01

Updated: 2026-10-21

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 80/100

The gist: The gist The proposed framework successfully captures sharp shock discontinuities across a wide range of supersonic and hypersonic flows (2 ≤ Ma ≤ 15) around a circular cylinder without relying

Key concepts

Hybrid Convolutional Architecture: The network uses two types of convolutions: a 1D radial convolution to capture long-range upstream/downstream dependencies and an anisotropic 2D azimuthal convolution. This design embeds a directional bias, allowing the network to process the flow's physics by separating radial wave propagation from tangential grid topology, mimicking finite-difference operators inherent in fluid dynamics.
Regime-Dependent Dynamic Residual Scaling: To handle extreme stiffness at different speeds, the loss function scales residuals based on Mach number. For high Mach numbers (Ma>=3), residuals are scaled down to reduce gradient stiffness. Conversely, for low supersonic flows (Ma=2), residuals are multiplied by Mach number powers to counteract spectral bias and force the network to resolve weak shock discontinuities.
Total Variation Loss (Ltv): This regularization term is added specifically in the azimuthal direction to suppress non-physical artifacts like the carbuncle phenomenon near stagnation lines. It acts as a smoothing penalty that helps stabilize optimization, although excessive application at high Mach numbers can cause some smearing of shock fronts.
Mach-Number Guided Dynamic Scaling: This strategy dynamically adjusts how the network learns by changing the weighting of physical constraints based on flow speed. It is crucial for balancing stability across disparate regimes, ensuring that the optimization process remains viable whether solving a low-speed supersonic problem or a high-speed hypersonic shock structure.

Terminology

Fidelity — 1× in this paper · explained in 2 episodes
Hybrid Architecture — 1× in this paper · explained in 2 episodes
Physics-Informed Neural Network — 1× in this paper
Physics-Informed Neural Network (PINN) — 1× in this paper · explained in 3 episodes
Spectral Bias — 1× in this paper · explained in 2 episodes

Summary

The gist The proposed framework successfully captures sharp shock discontinuities across a wide range of supersonic and hypersonic flows (2 ≤ Ma ≤ 15) around a circular cylinder without relying on referential data.

How it works

The core innovation is a fully data-free Physics-Informed Neural Network (PINN) capable of solving compressible inviscid flows ranging from supersonic to hypersonic, up to Ma=15 around a circular cylinder The architecture addresses the spatial blindness of standard Multi-Layer Perceptrons by combining radial 1D convolutions with anisotropic azimuthal 2D convolutions to embed directional inductive biases To stabilize optimization across disparate flow regimes, a regime-dependent, Mach-number-guided dynamic residual scaling strategy is introduced which scales down residuals in high-Mach regimes to mitigate extreme gradient stiffness and applies penalty multipliers in low-supersonic flows to overcome spectral bias

Architectural Design and Inductive Biases

The network design philosophy establishes a directional inductive bias tailored to the physics of the polar Euler equations, decoupling spatial processing into two orthogonal operations

The specific components of the hybrid architecture include:

A large-kernel 1D convolution along the radial axis to capture long-range upstreamdownstream inter-dependencies, allowing the network to anchor the detached bow shock between the free-stream and the wall
An anisotropic 2D convolution along the azimuthal axis, preserving the continuous (r, θ) grid topology to mathematically mimic tangential finite-difference operators inherent to governing PDEs

Loss Function Formulation and Stabilization

The training objective minimizes a composite loss function Ltotal which combines globally evaluated physical constraints with region-specific penalty terms The key components of the loss function include:

**) Euler Residual Loss (Lpde): This is governed by a Mach-number-based dynamic residual scaling where momentum and energy residuals are weighted with Ma−2∞ and Ma−4∞ for 3 ≤ Ma∞ (PINN type A) or Ma2∞ and Ma4∞ respectively for Ma∞ ≤ 2 (PINN type B) The paper notes that the Hessian matrix of the loss function exhibits a quartic dependence on the Mach number, kHk ∼ O(Ma4∞), explaining why optimization landscapes become extraordinarily stiff in high-Mach regimes 5) Enthalpy Loss (LH): This purely algebraic relation provides a direct, stable gradient signal to anchor the thermodynamic state without the noise of higherorder derivatives 6) Nose Region Loss (Lnose): Evaluated specifically within the azimuthal range θ ∈ [0.9π, π], this penalty acts as a high-resolution magnifying glass ensuring strict enforcement of the Rankine-Hugoniot jump and correct shock detachment distance 7) Boundary and Stagnation Anchor Losses (Lin,Lwall,Lsym,Lstag): These terms provide critical Dirichlet and Neumann anchors that guide the global optimization trajectory 8) Upstream Fixing Mask Loss (Lmask): This enforces strictly uniform freestream conditions within the upstream region to prevent non-physical pre-shock oscillations 9) Total Variation Loss (Ltv): An azimuthal regularization term introduced to explicitly suppress the non-physical carbuncle phenomenon near the stagnation line The static weighting strategy for these losses is rigorously defined based on physical prioritization, with λmask being set to 1000 in certain cases 15) Optimization Strategy and Training Protocol: A curriculum learning strategy and a twostage hybrid optimization are employed, starting with the AdamW optimizer for global exploration followed by a fine-tuning phase using the L-BFGS algorithm 21) The final converged loss settles at progressively higher values in extreme hypersonic regimes because of the heavily amplified Total Variation (TV) loss penalty, reflecting a structural trade-off between macroscopic stability and shock suppression 25) The regime-dependent nature of optimal loss balancing is demonstrated by comparing PINN type A (scaling down for Ma∞ ≥ 3) and PINN type B (scaling up for Ma∞ = 2), showing that multiplication of PDE residuals by powers of the Mach number is imperative to overcome spectral bias in low-supersonic flows 24) The analysis shows that the L-BFGS optimization phase is remarkably short because the AdamW optimizer has already navigated the network into a highly stiff and extremely narrow local plateau, which is characteristic of the hyperbolic Euler equations with shock discontinuities 23) The final converged loss settles at progressively higher values in extreme hypersonic regimes because of the heavily amplified Total Variation (TV) loss penalty, reflecting a structural trade-off between macroscopic stability and shock suppression 25) The most significant and novel finding of this study is that the optimal loss balancing strategy in compressible PINNs is fundamentally regime-dependent 24) The integration of artificial viscosity, Total Variation (TV) loss, and a novel Upstream Fixing boundary loss proved highly effective in achieving global convergence during the two-stage optimization process 25) The mathematical smoothing penalties required to stabilize the optimization – particularly at the extreme Ma∞ = 15 condition, where the TV loss weight was amplified by a factor of 10 – inevitably smear the shock front and induce non-linear discrepancies in variables such as the velocity profile along the stagnation streamline 27) The paper concludes that while the proposed method may not yet achieve the exact, infinitely sharp shock resolution of traditional Computational Fluid Dynamics (CFD) schemes, it demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments 25) The integration of Mach-numberguided dynamic scaling and the hybrid convolutional architecture presented in this study establishes a crucial mathematical and physical foundation for the future practical application of PINNs to more complex compressible flow problems. 25) The most significant and novel finding of this study is that the optimal loss balancing strategy in compressible PINNs is fundamentally regime-dependent 24) The integration of artificial viscosity, Total Variation (TV) loss, and a novel Upstream Fixing boundary loss proved highly effective in achieving global convergence during the two-stage optimization process 25) The mathematical smoothing penalties required to stabilize the optimization – particularly at the extreme Ma∞ = 15 condition, where the TV loss weight was amplified by a factor of 10 – inevitably smear the shock front and induce non-linear discrepancies in variables such as the velocity profile along the stagnation streamline 27) The paper concludes that while the proposed method may not yet achieve the exact, infinitely sharp shock resolution of traditional Computational Fluid Dynamics (CFD) schemes, it demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments. 25) The integration of Mach-numberguided dynamic scaling and the hybrid convolutional architecture presented in this study establishes a crucial mathematical and physical foundation for the future practical application of PINNs to more complex compressible flow problems. 25) The most significant and novel finding of this study is that the optimal loss balancing strategy in compressible PINNs is fundamentally regime-dependent 24) The integration of artificial viscosity, Total Variation (TV) loss, and a novel Upstream Fixing boundary loss proved highly effective in achieving global convergence during the two-stage optimization process 25) The mathematical smoothing penalties required to stabilize the optimization – particularly at the extreme Ma∞ = 15 condition, where the TV loss weight was amplified by a factor of 10 – inevitably smear the shock front and induce non-linear discrepancies in variables such as the velocity profile along the stagnation streamline 27) The paper concludes that while the proposed method may not yet achieve the exact, infinitely sharp shock resolution of traditional Computational Fluid Dynamics (CFD) schemes, it demonstrates unprecedented optimization stability and physical validity for data-free PINNs in extreme aerodynamic environments. 25) The integration of Mach-numberguided dynamic scaling and the hybrid convolutional architecture presented in this study establishes a crucial mathematical and physical foundation for the future practical application of PINNs to more complex compressible flow problems.

Improvements for AI systems

Spatial Inductive Bias for Flow Resolution: The hybrid radial 1D convolution and anisotropic azimuthal 2D convolution embed a directional inductive bias directly into the neural topology to address spatial blindness. This allows the system to capture directional characteristics of supersonic flow, where the radial direction is dominated by hyperbolic wave propagation and sharp discontinuities, enabling it to localize shock interfaces accurately.
Regime-Dependent Loss Scaling: The Mach-number-guided dynamic residual scaling strategy mitigates gradient stiffness across different flow speeds. For high-Mach flows (Ma >= 3), residuals are scaled down to mitigate extreme gradient stiffness, while for low-supersonic flows (Ma=2), residuals are multiplied to overcome the network’s spectral bias, forcing it to explicitly resolve weak shock discontinuities.
Stability Enhancement via Physical Constraints: The loss formulation incorporates several novel terms that stabilize optimization. This includes an exact analytical solution at the stagnation point to establish a global thermodynamic anchor and a Total Variation (TV) loss to explicitly suppress upstream noise and the non-physical carbuncle phenomenon.
Dynamic Weighting Heuristic for Optimization: Instead of adaptive weighting schemes that fail for hyperbolic equations, a static, physicsguided heuristic is used. This sets weights based on physical prioritization, such as assigning dominant weights (lambda 10 squared - 10 3) relative to the scaled PDE loss (lambda pde = 1) for boundary conditions.
Enhanced Physical Fidelity in High-Mach Flows: The system can predict macroscopic flow structures across a wide range of supersonic and hypersonic regimes (Ma=2 to Ma=15). However, it will produce non-linear discrepancies in velocity profiles for Ma >= 9 due to the excessive smoothing effect of artificial viscosity and TV loss, which is an inherent trade-off for optimization stability.
Low-Supersonic Fidelity: For low Mach numbers (Ma=2), the system can accurately approximate reference CFD solutions by multiplying PDE residuals by Mach number powers, preventing it from smoothing out the weak discontinuity and ensuring it strictly adhere[s] to physical fidelity.

Abstract

A data-free finite-volume convolutional neural network (FV-CNN) is formulated for steady Euler flow around a cylinder at eight Mach numbers, 2--15. Shared-face fluxes define conservative spatial balances, while automatic differentiation supplies parameter gradients. The derivative mismatch in the earlier automatic-differentiation-based convolutional neural network (AD-CNN) is derived. No computational fluid dynamics (CFD) or experimental field targets enter training; reference-independent selection of coefficients, seeds, and checkpoints is not established. Billig-based upstream fixing conditions shock-location agreement.For eight adopted fields, density and pressure relative L 2 discrepancies from finite-resolution CFD are 5.43 -- 18.69% and 6.25 -- 19.70%. Closed-band defects are smaller than AD-CNN in 32/32 self-centered and 22--31/32 CFD-centered comparisons over seven half-widths, 0.06--0.12 cylinder diameters. AD-CNN retains smaller enthalpy root-mean-square (RMS) discrepancies in 7/8 cases, shock-position RMS at Mach 11 and 15, and temperature and transverse-velocity discrepancies at Mach 3.Ten new Mach-15 initializations and adopted seed 42 yield density discrepancies of 12.04 -- 43.70% (mean 24.34%, sample standard deviation 11.78 percentage points); the new ten alone give 25.04% and 12.17 points. Dispersion exceeds the historical adopted-field gap of 3.40 points; six of eleven outcomes are below AD-CNN. Transition-width ratios of AD-CNN to FV-CNN remain 4.07 -- 4.66, and all 44 self-centered seed--component balances at half-width 0.10 cylinder diameters are smaller for FV-CNN. Local sharpness and this balance metric persist, while initialization-robust accuracy superiority is not established.Component interventions remain one-seed conditional responses; Mach 2 uses a separate budget-limited protocol.