Global Convergence Proof for DGM and PINNs: A Long-Awaited Mathematical Guarantee

Forum topic · 小凯 · 2026-07-28

Summary

A forum post discusses a new paper by Justin Sirignano, Konstantinos Spiliopoulos, and Samuel Cohen titled 'Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs' (arXiv:2607.24726). For nearly a decade, Physics-Informed Neural Networks (PINNs) and the Deep Galerkin Method (DGM) have been widely used to solve partial differential equations, but their training was known to be unstable, and no guarantee existed that gradient descent would avoid bad local minima. The paper provides a rigorous mathematical proof that, for a class of semilinear PDEs, over-parameterized neural networks trained by gradient descent converge globally to the true PDE solution. The proof combines the universal approximation theorem, neural tangent kernel (NTK) analysis showing that wide networks yield smooth loss landscapes, and dynamical systems stability theory. The post explains why semilinear PDEs matter (reaction-diffusion, population dynamics, neural signaling), why traditional grid-based methods suffer from the curse of dimensionality, and why the result boosts confidence in scientific machine learning—while noting that quasilinear and fully nonlinear PDEs like Navier-Stokes remain open problems.

Paper Information

Title: Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs Authors: Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen arXiv: 2607.24726 Categories: cs.LG, math.NA Posted: 2026-07-27

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The Problem: PINN Training Gets Stuck

Imagine a physicist simulating fluid flow in a pipe using the Navier-Stokes equations. Classical numerical methods (finite elements, finite differences) are mature but become extremely expensive for complex geometries and high-dimensional problems. Physics-Informed Neural Networks (PINNs) promise an attractive alternative: train a neural network to satisfy the PDE directly, with no grid required.

In practice, however, training is unstable: the loss decreases, then plateaus or oscillates. A low loss does not guarantee an accurate solution. The reason is fundamental—neural network loss functions are non-convex, so gradient descent can get trapped in local minima that are not solutions of the PDE. This problem has bothered the scientific machine learning community for nearly ten years. The paper discussed here finally provides an affirmative answer for an important class of equations.

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Background: DGM and PINN

Two closely related methods emerged in 2018–2019:

Deep Galerkin Method (DGM): approximate the solution with a deep network u_θ(x, t) and minimize the PDE residual loss:
PINNs: add boundary and initial condition terms:

If L(θ) = 0, the network exactly satisfies the PDE. But without convergence guarantees, gradient descent could stop anywhere.

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The Main Result

The paper proves global convergence for a class of semilinear PDEs (nonlinear in the solution and its first derivatives, but linear in the highest-order derivatives—e.g., reaction-diffusion equations u_t = Δu + f(u)). Specifically: gradient descent training of a neural network on the PDE residual loss converges to the true solution of the PDE.

Key ingredients of the proof:

1. Universal approximation: some network parameters θ* exist such that u_θ* approximates the true solution u* arbitrarily well. 2. Loss landscape structure: the structure of semilinear PDEs (uniqueness, regularity of solutions) ensures gradient descent does not get trapped in "bad" local minima. 3. Over-parameterization: sufficiently wide networks make the loss landscape smoother, and gradient descent converges to the global optimum with high probability.

#### Technical perspective

Neural Tangent Kernel (NTK): when the network is very wide, training dynamics are governed by a kernel; the loss landscape becomes nearly smooth, so gradient descent almost surely reaches the minimum—which corresponds to the true PDE solution.
Dynamical systems stability: the training dynamics are "attracted" toward the neighborhood of the correct solution, in a way compatible with the attractor structure of the PDE's solution space.

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Why It Matters

1. Theory finally catches up: PINNs and DGM have been applied across fluid mechanics, solid mechanics, electromagnetics, and quantum mechanics without theoretical guarantees. This result shows that for semilinear PDEs, a sufficiently wide network trained long enough is guaranteed to converge to the correct solution. 2. A first step toward harder PDEs: quasilinear PDEs (like Navier-Stokes) and fully nonlinear PDEs remain open challenges. As with many big mathematical problems, this is an important first step. 3. Confidence for SciML: the result gives a strong signal that "black box + physics constraints" is reliable under certain conditions.

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Conclusion

Back to our physicist: for "friendly" semilinear PDEs, the fear of wasting time in a local minimum can finally be put to rest. Practical challenges remain—training efficiency, architecture design, hyperparameter tuning—but the foundational guarantee now exists.

> "If you can't explain it to a six-year-old, you don't really understand it yourself." — Richard Feynman

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Reference Sirignano, J., Spiliopoulos, K., & Cohen, S. (2026). Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs. arXiv:2607.24726.