AI4PDE

Models in motion

Research samples

Representative applications of energy-based learning, physics-informed deep collocation, transfer learning, and stochastic computation in Xiaoying Zhuang's research.

01 / Variational PDE learning

Energy approach to computational mechanics

A mechanics-informed framework uses variational energy principles to define neural-network objectives for partial differential equations.

  • Connects weak and variational formulations with machine learning
  • Provides concepts, implementation details, and benchmark applications
  • Uses physical energy rather than labelled solutions as the central objective
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From mechanics to optimization
PDE→Energy→Network→uθ
Π(u) = ∫Ω Ψ(ε(u))dΩ − ∫Γt t̄·u dΓ

02 / Thin-plate mechanics

Deep collocation for Kirchhoff plates

A deep collocation method solves fourth-order Kirchhoff plate bending by embedding the governing equation and boundary conditions into the training loss.

  • Mesh-free neural approximation of plate deflection
  • Physics and boundary constraints enforced at collocation points
  • Demonstrates deep learning for higher-order structural PDEs
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Kirchhoff plate equation D∇⁴w = q

Approximate deflection while satisfying the fourth-order governing equation.

03 / Non-homogeneous media

3D potential problems with material transfer

Physics-informed deep collocation addresses three-dimensional potential equations with spatially varying material properties, supported by material transfer learning and global sensitivity analysis.

  • Tests activation, sampling, and optimizer configurations
  • Transfers across material gradations and parameters
  • Includes global sensitivity analysis and convergence analysis
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Non-homogeneous potential field ∇ · [k(x)∇uθ(x)] = f(x)

04 / Stochastic porous media

Architecture-searched deep collocation

A stochastic DCM combines sensitivity analysis, neural architecture search, and transfer learning for highly heterogeneous three-dimensional groundwater flow.

  • Sensitivity analysis narrows the hyperparameter search space
  • Strong architectures supply weights for efficient fine-tuning
  • Validated across dimensions with manufactured solutions
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Stochastic groundwater flowK(x, ξ) → hθ(x, ξ)

Learn pressure-head responses across heterogeneous conductivity fields.