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MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network in Applications to Multidimensional Coupled Burgers' Equations with High Reynolds Numbers

This paper introduces a MATLAB-based layerwise self-adaptive physics-informed neural network that utilizes dynamic weighting and dual-phase optimization to accurately simulate multidimensional coupled Burgers' equations with high Reynolds numbers, effectively overcoming the stability and mesh-dependency limitations of conventional methods in capturing sharp shock fronts.

Original authors: Harish P. Bhatt, Xi Chen, Jingsai Liang

Published 2026-06-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Harish P. Bhatt, Xi Chen, Jingsai Liang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a drop of ink spreads through a glass of water, but with a twist: the water is moving incredibly fast, and the ink doesn't just diffuse smoothly; it forms a razor-sharp, jagged line (a "shock front") that moves and changes shape. This is the kind of problem scientists face when simulating fluids in high-speed scenarios, described mathematically by something called the Burgers' equations.

For a long time, computers have struggled to draw these sharp lines accurately. Traditional methods are like trying to draw a jagged lightning bolt using only a thick, blunt crayon on a grid paper. If the grid is too coarse, the line looks blocky and wrong. If you make the grid fine enough to capture the sharpness, the computer gets overwhelmed and crashes because there are too many grid squares to calculate.

The New Approach: A Smart, Self-Correcting Painter

This paper introduces a new "painter" called LSAAL-PINN. Instead of using a grid, this painter is a Neural Network (a type of AI) that learns the rules of physics directly. Think of it not as a grid, but as a continuous, flexible sheet that can stretch and bend to fit the shape of the ink perfectly.

However, standard AI painters often get confused. They might focus too much on getting the starting position right and forget the rules of how the fluid moves, or vice versa. To fix this, the authors gave their AI painter two special superpowers:

1. The "Self-Adjusting Volume Knobs" (Layerwise Self-Adaptive Weighting)

Imagine you are mixing a song with three different instruments: the Physics (the rules of fluid motion), the Initial Conditions (where the ink starts), and the Boundary Conditions (where the walls of the glass are).

Old AI: The volume knobs for these three instruments are fixed. If the "Physics" instrument is playing too loudly (hard to solve), it drowns out the other two, and the AI gets stuck.
New AI (LSAAL-PINN): This AI has smart, self-adjusting knobs. As it trains, it listens to the music. If the Physics part is too loud and causing a mess, it automatically turns that volume down and turns up the volume on the other parts to balance the mix. It does this constantly, ensuring no single rule dominates and confuses the AI.

2. The "Two-Phase Training" (Dual-Phase Optimization)

The AI learns in two distinct stages, like a student taking a test:

Phase 1 (The Rough Draft): The AI uses a fast, general-purpose optimizer (called Adam) to get a good "rough draft" of the solution. It's like sketching the general shape of the ink wave quickly.
Phase 2 (The Polish): Once the sketch is done, the AI switches to a super-precise, second-order optimizer (called L-BFGS). This is like taking a fine-tipped pen and carefully refining every curve and sharp edge. This phase ensures the final drawing is incredibly accurate, especially at those tricky, sharp shock fronts.

The Results: Drawing the Lightning Bolt

The authors tested this new method on 1D, 2D, and 3D versions of the fluid problem, including scenarios where the fluid moves very fast (high Reynolds numbers).

The Competition: They compared their new AI against two older versions:
1. A standard AI that only used the fast "Rough Draft" phase.
2. A standard AI that used both phases but didn't have the "Self-Adjusting Volume Knobs."
The Winner: The new LSAAL-PINN won every time.
- It drew the sharp, jagged shock fronts with much higher precision.
- It made fewer mistakes (lower error rates) than the other methods.
- It was able to handle the "stiff" and difficult math of high-speed fluids without getting stuck or producing wobbly, incorrect lines.

The Bottom Line

The paper claims that by combining smart, self-adjusting weights (to balance the learning rules) with a two-step training process (sketch then refine), this new MATLAB-based AI can solve complex fluid problems much better than previous methods. It successfully captures the sharp, sudden changes in fluid flow that usually break other computer models, all without needing a massive, expensive grid of data points.

Note: The paper focuses strictly on solving these specific mathematical equations (Burgers' equations) to test the AI's accuracy. It does not claim to have solved real-world engineering problems, medical issues, or climate models yet; it simply proves the method works exceptionally well on these specific mathematical test cases.

Technical Summary: MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network for Multidimensional Coupled Burgers' Equations

Problem Statement
The paper addresses the computational challenges associated with simulating the spatio-temporal solution profiles of multidimensional coupled Burgers' equations (MCBEs) at high Reynolds numbers. As time evolves, these equations generate sharp shock fronts and steep solution gradients. Conventional mesh-based numerical methods, such as finite differences and finite elements, struggle with these features due to poor stability, strong mesh dependency, and the computational prohibitions of the "curse of dimensionality" when extremely fine grids are required. Furthermore, standard Physics-Informed Neural Networks (PINNs) often fail to resolve these high-Reynolds-number problems because the dominant convective terms create a "stiff" loss landscape. This stiffness causes optimizers to stall in local minima, preventing the network from accurately capturing the sharp gradients characteristic of the solution.

Methodology
The authors propose an improved framework termed LSAAL-PINN (Layerwise Self-Adaptive PINN trained with Adam and L-BFGS). The methodology is implemented in MATLAB using the Deep Learning Toolbox for automatic differentiation and is designed to solve $d$ -dimensional CBEs ( $d=1, 2, 3$ ).

Key methodological components include:

Network Architecture: A Deep Neural Network (DNN) approximates the solution $u(x, t)$ . The network utilizes a feedforward structure with $k$ hidden layers, employing the hyperbolic tangent ( $\tanh$ ) activation function to ensure the smoothness required for computing higher-order derivatives via automatic differentiation.
Layerwise Self-Adaptive Weighting: Unlike standard PINNs that use fixed weights for different loss components, this framework employs a dynamic strategy. The total loss function $L(\theta, \lambda)$ is a weighted sum of the PDE residual ( $L_{PDE}$ ), initial condition residual ( $L_{IC}$ ), and boundary condition residual ( $L_{BC}$ ). The weights $\lambda = [\lambda_{PDE}, \lambda_{IC}, \lambda_{BC}]$ are trainable parameters updated via gradient ascent to maximize the total loss, ensuring no single residual term dominates the training process. These weights are mapped through a logarithmic transformation to maintain strict positivity and numerical stability.
Dual-Phase Optimization: The training process utilizes a two-stage strategy:
- Phase 1 (Global Optimization): The network parameters $\theta$ and adaptive weights $\lambda$ are updated iteratively using the Adam optimizer.
- Phase 2 (Refinement): The adaptive weights $\lambda$ are frozen, and the optimization transitions to the second-order L-BFGS algorithm to refine the solution and accelerate convergence to a prescribed tolerance.
Sampling: To effectively capture steep gradients, the framework uses Latin Hypercube Sampling (LHS) to generate collocation points, ensuring points are distributed near transition regions.

Key Contributions and Experimental Setup
The study introduces the LSAAL-PINN framework and validates its effectiveness by comparing it against two baseline variants:

AL-PINN: Standard PINN with Adam and L-BFGS optimizers but without self-adaptive weighting.
A-PINN: Standard PINN trained only with the Adam optimizer.

The authors conducted numerical experiments on 1D, 2D, and 3D coupled Burgers' equations with high Reynolds numbers (low viscosity $\nu$ ). The accuracy was measured using the relative $L_2$ -error norm.

Results

1D Burgers' Equation: The LSAAL-PINN achieved a relative $L_2$ -error of $1.79 \times 10^{-4}$ , outperforming AL-PINN ( $3.10 \times 10^{-4}$ ) and A-PINN ( $8.65 \times 10^{-3}$ ). The results demonstrated that LSAAL-PINN aligned most precisely with the exact solution, particularly at the shock front.
2D Coupled Burgers' Equations (High Reynolds Number): For $\nu = 0.0001$ , the LSAAL-PINN provided results approximately 8 times more accurate than AL-PINN and 1385 times more accurate than A-PINN for the velocity component $v$ at $t=1$ . The A-PINN failed to capture the shock-like discontinuity, while the LSAAL-PINN successfully resolved the steep gradients.
3D Coupled Burgers' Equations: The LSAAL-PINN consistently yielded the lowest $L_2$ -errors across various time levels compared to the other variants. The framework also successfully solved 3D CBEs with a Reynolds number of $Re=100$, maintaining accuracy in the magnitude of $10^{-3}$ .
Convergence Behavior: Loss plots indicated that while A-PINN stagnated or oscillated, the dual-phase optimization (Adam followed by L-BFGS) in both AL-PINN and LSAAL-PINN led to significant loss reduction. The addition of self-adaptive weighting in LSAAL-PINN further minimized the total loss compared to AL-PINN.

Significance and Claims
The paper claims that the integration of a layerwise self-adaptive weighting strategy with a dual-phase optimization approach is sufficient to capture steep gradient solution profiles in MCBEs without requiring computationally expensive pointwise weighting strategies. The authors assert that this framework is more effective, accurate, and robust than standard PINNs for simulating spatio-temporal dynamics in high-Reynolds-number flows.

The study highlights that while existing literature has largely focused on 1D Burgers' equations at moderate Reynolds numbers (often in Python), this work advances the field by successfully solving 2D and 3D CBEs with high Reynolds numbers using a MATLAB-based implementation. The authors conclude that the proposed framework effectively addresses the stiffness of the loss landscape, enabling accurate prediction of sharp shock fronts that standard methods struggle to resolve. Future work is suggested to extend this framework to other challenging time-dependent PDEs.