This paper studies P-trainable neural networks, where a proportion P of hidden neurons are trainable and the rest stay fixed as random features, bridging random feature methods and fully trainable networks. The authors establish two-layer approximation bounds showing that a small trainable fraction can capture most of the gain when initialization mismatch is large, yielding a phase transition in rates of the form $\mathcal{O}(((P/\kappa_1^*+(1-P)/\kappa_0^*)m/\log m)^{-\alpha})$. They verify these assumptions explicitly for spherical biased-ReLU features and present numerical experiments showing analogous phase transitions in training losses across architectures with up to 15 layers.

The paper presents structure-preserving reduced-order models for two high-speed aeroelastic challenge problems—a clamped panel at Mach 1.92 (RC-19) and a cantilever plate at Mach 5.85 (HyMAX). Because the experimental packages lack time-resolved responses or repeated runs, the author proposes a learned model that retains the exact physics-based potential energy, fits generalized aerodynamic and damping forces by least squares, and trains only a dissipation matrix engineered to remain positive definite. An exact discrete-gradient integrator then guarantees an energy balance to machine precision regardless of the trained parameters, aiming to show that physical structure, not large datasets, can carry predictive capability in chaotic flutter regimes.

This paper tackles certified neural approximation of the Dirichlet Monge--Amp\`ere equation, a fully nonlinear PDE whose solutions must be convex. The authors study implicit-depth equilibrium networks built from softplus units with nonnegative weights, proving that a resolvent--curvature cancellation improves the Hessian growth bound from $(1-q)^{-3}$ to the sharp exponent $(1-q)^{-2}$. They then derive residual-based a posteriori estimates and validate them with interval-arithmetic certificates on trained models, yielding rigorous uniform evaluation guarantees for implicit neural PDE solvers.

This paper develops a physics-informed Kolmogorov-Arnold network (KAN) solver for free-boundary PDEs, including elliptic obstacle problems and the time-dependent Stefan problem. The core idea is to replace fixed MLP node activations with learnable univariate B-splines on network edges, aiming for localized resolution of contact sets and moving interfaces. Accurate, parameter-efficient approximation of these coupled unknowns matters because classical mesh-based methods require explicit interface tracking or costly refinement, while standard PINNs may blur sharp free boundaries.

Physics-informed neural networks typically enforce boundary conditions via penalty terms, leading to approximate satisfaction and training pathologies. This paper proposes a systematic method to enforce Dirichlet, Neumann, and Robin conditions exactly on curved quadrilateral domains using Theory of Functional Connections (TFC) combined with transfinite interpolation. The key innovation is handling compatibility constraints at vertices where mixed boundary conditions meet, particularly when two Neumann/Robin boundaries intersect, by decomposing the problem into a four-step procedure.

This paper solves stability and bifurcation analysis for nonlinear PDEs using Physics-Informed Random Projection Neural Networks (PI-RPNNs). The core innovation is a matrix-free shift-invert Krylov-Arnoldi method operating directly in weight space to circumvent the exponential singular value decay of the random collocation matrix $\Psi$. This enables reliable computation of leading eigenpairs for detecting saddle-node, Hopf, and pitchfork bifurcations without requiring additional PDE solves beyond the initial training.

This paper extends In-Context Operator Networks (ICONs)—which learn PDE solution operators via in-context learning without retraining—to higher-order and higher-dimensional PDEs. The authors test on 19 problem types including the heat equation and 3D linear PDEs, finding that while point-wise accuracy degrades for complex OOD problems, the model retains qualitative solution behavior.