Physics-informed machine learning methods for nonlinear problems in structural mechanics
Μέθοδοι μηχανικής μάθησης με ενσωμάτωση φυσικής γνώσης για την επίλυση μη γραμμικών προβλημάτων στη μηχανική
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Keywords
Weighted Sobolev spaces ; Demokritos ; Scientific computing ; Hardy inequality ; Physics-informed neural networks ; Composite laminates ; Muckenhoupt weights ; Adaptive collocation ; B11 coupling ; FSDT ; FEM ; Singular perturbation ; PINN ; Coercivity ; Nash implicit function theoremAbstract
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.