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Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp

Dibyakanti Kumar, Anirbit Mukherjee

TL;DR

The paper addresses solving Burgers' PDE near finite-time blow-up with Physics-Informed Neural Networks (PINNs) by deriving rigorous generalization bounds. It develops two key results: a $(d+1)$-dimensional Burgers' bound that controls the $L^2$-risk via PDE, initial, and boundary residuals, and a 1+1D blow-up bound that remains stable as the solution approaches singularity; both bounds are computable from surrogate residuals and boundary data and do not rely on the training loss details. Through experiments on 1D and 2D Burgers' blow-up scenarios, the authors show strong correlation between the predicted bounds and the actual $L^2$-error, with correlation improving for wider networks and training time remaining roughly constant near blow-up. These results provide a theoretical lens for evaluating PINN reliability in singular regimes and suggest directions for refining PINN architectures and bounds for higher-dimensional blow-ups in fluid-like PDEs.

Abstract

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the $\ell_2$-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp

TL;DR

The paper addresses solving Burgers' PDE near finite-time blow-up with Physics-Informed Neural Networks (PINNs) by deriving rigorous generalization bounds. It develops two key results: a -dimensional Burgers' bound that controls the -risk via PDE, initial, and boundary residuals, and a 1+1D blow-up bound that remains stable as the solution approaches singularity; both bounds are computable from surrogate residuals and boundary data and do not rely on the training loss details. Through experiments on 1D and 2D Burgers' blow-up scenarios, the authors show strong correlation between the predicted bounds and the actual -error, with correlation improving for wider networks and training time remaining roughly constant near blow-up. These results provide a theoretical lens for evaluating PINN reliability in singular regimes and suggest directions for refining PINN architectures and bounds for higher-dimensional blow-ups in fluid-like PDEs.

Abstract

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the -distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.
Paper Structure (20 sections, 4 theorems, 66 equations, 5 figures)

This paper contains 20 sections, 4 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Informal Summary of Our Results
  4. Related Works
  5. Inapplicability of Classical Numerical Analysis Error bounds to PINN Experiments
  6. Results
  7. Generalization Bounds for the $(d+1)$-Dimensional Burgers' PDE
  8. Generalization Bounds for a Finite-Time Blow-Up Scenario with (1+1)-Dimensional Burgers' PDE
  9. Experiments
  10. The Finite-Time Blow-Up Case of (1+1)-Dimensional Burgers' PDE from Section \ref{['subsec:1d-burgers']}
  11. Testing Against a (2+1)-Dimensional Exact Burgers' Solution with Finite-Time Blow-Up
  12. Conclusion
  13. Proofs for the Main Theorems
  14. Proof of Theorem \ref{['th:ndburgers.error_upperbound']}
  15. Useful Lemmas
  16. ...and 5 more sections

Key Result

Theorem 4.1

Let $d \in {\mathbb N}$ and ${\bm{u}} \in C^1(D \times [t_0, T] )$ be the unique solution of the (d+1)-dimensional Burgers' equation given in equation ndburgers.2. Then for any $C^1$ surrogate solution to equation ndburgers.2, say ${\bm{u}}_\theta$, the $L^2$-risk with respect to the true solution i where,

Figures (5)

  • Figure 1: A demonstration of the visual resemblance between the neurally derived solution for equation \ref{['eq:burger_pdes:6']} (left) and the true solution (right) at different values of the $\delta$ parameter getting close to the PDE with blow-up at $\delta=1$. A PINN with a width of $300$ and a depth of $6$ was trained to generate the plots on the left.
  • Figure 2: Demonstration of the presence of high correlation between the LHS (the true risk) and the RHS (and the derived bound) of equation (\ref{['eq:int_burger_bound_th_1']}) in Theorem \ref{['th:int_burger_gen_error_bound']} over PDE setups increasingly close to the singularity. Each experiment is labeled with the value of $\delta$ in the setup of equation \ref{['eq:burger_pdes:6']} that it corresponds to.
  • Figure 3: These plots show the behaviour of LHS (the true risk) and RHS (the derived bound) of equation (\ref{['ndburgers.upperbound']}) in Theorem \ref{['th:ndburgers.error_upperbound']} for different values of the $\delta$ parameter that quantifies proximity to the blow-up point. In the left plot each point is marked with the value of the $\delta$ at which the experiment is done and the right figure, for clarity, this is marked only for experiments at $\delta > \frac{1}{2}$.
  • Figure 4: The above plot tracks the RHS of equation (\ref{['eq:int_burger_bound_th_1']}) in Theorem \ref{['th:int_burger_gen_error_bound']} for training a depth $2$ net at different widths towards solving equation \ref{['eq:burger_pdes:6']} at $\delta = \frac{1}{2}$
  • Figure 5: The above plots show that the time taken to train a PINN on equation \ref{['eq:burger_pdes:6']} barely changes for different values of $\delta$ - a measure of proximity to blow-up and that this holds at two widely separated widths of the net.

Theorems & Definitions (7)

  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Lemma A1
  • proof
  • Theorem A2
  • proof