Neural Operators for Stability Analysis
bravo_abad · x · 2026-07-17
This article introduces a method advancing Neural Operators from mere "trajectory prediction" tools to analyzing "stability and bifurcations."
Core Concept
Instead of treating neural operators purely as forward simulators, the author trains them as short-time-step maps combined with classical numerical analysis:
- Using Newton-GMRES to solve fixed-point equations u = S[u] directly
- Using Arnoldi to compute the dominant eigenvalues of the Jacobian matrix to assess stability
- Using arc-length continuation to generate bifurcation diagrams
The advantage is that it can locate stable equilibria and unstable equilibria that are difficult for traditional long-time simulations to reach.
Methodology Highlights
- Matrix-free: Avoids explicitly constructing a dense Jacobian matrix, obtaining its action on a vector via finite differences.
- Local modeling: Can be trained locally in both time and space using smaller input blocks (teeth) and inactive gaps, reducing training difficulty, lowering data requirements, and improving solve conditions.
- RandONets: A random shallow neural operator where training reduces to regularized linear least squares. These regularizers act as spectral filters, suppressing high-frequency noise and keeping the learned Jacobian spectrum reliable.
Experimental Results
The method was validated on three benchmarks:
- Pitchfork bifurcation in Allen-Cahn
- Saddle-node tipping point in Bratu (recovering accuracy up to 10^-5)
- Hopf bifurcation in FitzHugh-Nagumo
Results show that both bifurcation diagrams and eigenvalues align with reference solvers, and the time step size can be up to 200 times larger than reference schemes.
Significance
This approach is ideal for scenarios with sparse data and no reliable simulator, such as:
- Process and reactor engineering
- Energy systems
- Phase change materials
It elevates neural operators from "providing a prediction" to "helping you find where a system stabilizes and where it flips."
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