Summary
Standard machine-learning optimizers — Adam, L-BFGS, and their variants — treat the parameter space of a neural network as a flat Euclidean manifold, updating parameters along the negative gradient of a scalar loss. This thesis argues that this geometry is inadequate for a class of physics-informed machine learning (PIML) problems — specifically, singularly perturbed multiphysics PDEs with block-triangular structure — and proposes two connected remedies grounded in classical mathematical analysis. The claims are validated on a doubly-clamped [0◦/90◦] CFRP composite beam with Timoshenko FSDT kinematics and a shear parameter Πs = 206,897, which serves as a worst-case benchmark for standard PIML formulations. The first contribution is a formulation of Nash Implicit Function Theorem (IFT) Optimization: a structured decomposition of the PIML training problem into sequentially solvable subproblems, each operating in the natural function space of its governing PDE. When a multiphysics system has block-triangular PDE structure — as do the mixed ( ˆW , ˆM ) equations governing a doubly-clamped composite beam — the Nash decomposition allows the independent subsystem (bending moment ˆM ) to be converged first, providing a high-quality frozen field for the coupled subsystem (deflection ˆW ) in the next phase. Experiments confirm that this splitnetwork Nash PINN (M4) matches the accuracy of the baseline Mixed PINN (0.522% deflection error) while providing a principled convergence certificate absent from standard joint training. The second contribution is the use of Hardy inequality coercivity to repair the loss landscape of PINNs in the singularly perturbed regime Πs = A55L2/D11 ≫ 1. Standard three-field PINNs fail catastrophically (98.7% error) because the loss functional is non-coercive: the shear residual γxz ∼ Π−1 s ≈ 5×10−6 is seven orders of magnitude smaller than the rotation φ ∼ 0.13 rad, making the trivial solution a near-zero of the loss. The Hardy–Poincaré inequality, applied through curvature-adaptive collocation weights ωi = | ˆW ′′(ξi)|2 + η, restores weighted coercivity and reduces the loss plateau from 5×10−4 (uniform collocation) to 7×10−5 (Hardy-adaptive) — a 7-fold improvement confirmed experimentally. Both contributions are developed within a unified Hardy–Nash Framework: the Hardy inequality provides the correct function-space geometry (weighted Sobolev space W1,2 w (Ωh)); the Nash IFT provides the correct optimization architecture (block-decomposed, preconditioned gradient flow). The primary purpose of this work is to validate the effectiveness of the Hardy–Nash methodology on a well-characterised, low-dimensional benchmark — the stiffness problem of a doubly-clamped Timoshenko beam — before extending to higher-dimensional and more complex structural problems. All results are validated on this test case: a [0◦/90◦] CFRP composite beam with Timoshenko FSDT kinematics, von Kármán nonlinearity, and Πs = 206 897 — using a verification suite of eight independent checks (V1–V8) all of which pass.
