Zuazua E (2026)
Publication Language: English
Publication Type: Journal article, Online publication
Publication year: 2026
Book Volume: 7
Journal Issue: 59
Open Access Link: https://www.siam.org/publications/siam-news/articles/good-solutions-bad-coordinates-the-coercivity-gap-in-neural-partial-differential-equation-solvers/
Sometimes, a neural partial differential equation (PDE) solver may appear to be misbehaving, in that some weights grow without apparent bound as neurons collapse onto one another. In many numerical settings, such behavior would seem like a warning sign of instability, poor conditioning, or impending failure. In neural approximation, however, the diagnosis can be subtler: the coordinates may deteriorate while the computed physical field remains accurate.
This distinction matters because while parameter growth is often interpreted as numerical failure, in some neural PDE solvers it may instead signal that the architecture is approaching a legitimate state-space limit that lies outside of its finite parametrization. It matters even more as scientific machine learning (ML) becomes part of the numerical toolbox for PDEs and ordinary differential equations. Physics-informed neural networks (PINNs), deep Ritz methods, operator learning, and related techniques use trainable functions to encode physical laws, data, boundary conditions, and variational principles; representative entry points are the deep Ritz method [1] and the broader physics-informed ML framework [2]. These methods are especially appealing in high-dimensional settings, where classical grids can become prohibitively expensive.
The central question is familiar to numerical analysts but newly delicate in this setting: What exactly must converge? The optimizer evolves weights, biases, centers, and other parameters, but the differential equation only sees the realized physical state. The coercivity gap [4] names the separation between these two worlds; a neural PDE solver can produce states that converge even while the parameters that generate them diverge. Figure 1 summarizes this basic tension where the optimizer moves in parameter space, while the PDE judges the realized state.
APA:
Zuazua, E. (2026). Good Solutions, Bad Coordinates: The Coercivity Gap in Neural Partial Differential Equation Solvers. SIAM News, 7(59).
MLA:
Zuazua, Enrique. "Good Solutions, Bad Coordinates: The Coercivity Gap in Neural Partial Differential Equation Solvers." SIAM News 7.59 (2026).
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