Methods · operators · mechanics

Research areas

Xiaoying Zhuang's research with collaborators connects variational mechanics, physics-informed deep collocation, transfer learning, and stochastic computation across PDEs, heterogeneous materials, and complex geometries.

01 / Variational mechanics

Energy-based machine learning

The energy approach embeds variational principles directly into neural-network training. Instead of approximating a PDE only through pointwise residuals, it minimizes a physically meaningful functional and provides a systematic route from concepts to implementation and computational-mechanics applications.

Variational PDEsEnergy minimizationMechanics

Potential energy functionalΠ(uθ) = ∫Ω W(ε(uθ))dΩ − ∫Γt t̄·uθdΓ

Optimize a physically meaningful functional rather than labels alone.

02 / Physics-informed collocation

Deep collocation for PDEs

Deep collocation enforces governing equations and boundary conditions at sampled points. The framework has been developed for Kirchhoff plate bending, three-dimensional potential problems, porous media, and heat transfer in functionally graded materials.

Strong-form physicsPDE and boundary residuals provide supervision.

Mesh-free trainingCollocation points resolve irregular domains.

Collocation objectiveJ(θ) = Σᵢ ‖𝓛uθ(xᵢ) − f(xᵢ)‖² + J∂Ω

Train on equations and boundary conditions at sampled locations.

03 / Adaptation

Material transfer & domain adaptation

Transfer learning reuses trained parameters as materials, property gradients, or geometric domains change. Material transfer improves robustness for non-homogeneous media, while domain adaptation combines NURBS geometry with enhanced collocation for nonlinear PDEs.

Fine-tuningMaterial gradientsComplex geometry

Transfer pipelineθsource → fine-tune(Dtarget) → θtarget

Reuse physical features instead of retraining every problem from scratch.

04 / Uncertainty

Stochastic media & architecture search

For heterogeneous porous media, sensitivity analysis first identifies influential network choices. Neural architecture search then selects effective configurations, and transfer learning reduces the cost of stochastic three-dimensional groundwater-flow simulation.

Sensitivity analysisFocus search on influential hyperparameters.

Architecture searchReuse strong candidates through fine-tuning.

Stochastic responseK(x, ξ) → uθ(x, ξ)

Resolve flow responses across heterogeneous material realizations.

Key applications

Plates & potential fields

Applications include Kirchhoff plate bending and three-dimensional potential problems in non-homogeneous media.

Porous & stochastic media

Architecture search and transfer learning support stochastic groundwater-flow simulation in highly heterogeneous aquifers.

Transient heat transfer

Physics-informed time integration addresses functionally graded solids, irregular shapes, varied boundary conditions, and melting processes.