AI4PDE

Machine learning · physics · mechanics

Machine learning research

Xiaoying Zhuang's machine-learning research with collaborators develops neural representations, physics-informed objectives, transfer strategies, and architecture-search methods for PDEs, heterogeneous materials, and complex geometries.

01 / Physics-informed learning

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 functional
\[ \begin{aligned} \Pi[\boldsymbol{u}_\theta] &= \int_{\Omega} W\!\left(\boldsymbol{\varepsilon}(\boldsymbol{u}_\theta)\right)\, \mathrm{d}\Omega - \int_{\Gamma_t} \bar{\boldsymbol{t}}\!\cdot\!\boldsymbol{u}_\theta\,\mathrm{d}\Gamma,\\ \theta^\star &= \underset{\theta}{\operatorname{arg\,min}}\;\Pi[\boldsymbol{u}_\theta]. \end{aligned} \]

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 objective
\[ \begin{aligned} \mathcal{J}(\theta) ={}& \frac{1}{N_\Omega}\sum_{i=1}^{N_\Omega} \left\|\mathcal{L}u_\theta(\boldsymbol{x}_i)-f(\boldsymbol{x}_i)\right\|_2^2\\ &+\lambda_D\,\mathcal{J}_{\Gamma_D}(\theta) +\lambda_N\,\mathcal{J}_{\Gamma_N}(\theta). \end{aligned} \]

Train on equations and boundary conditions at sampled locations.

03 / Adaptation

Material transfer & sensitivity analysis

Transfer learning reuses trained parameters as materials, property gradients, or geometric configurations change. Material transfer improves robustness for non-homogeneous media, while global sensitivity analysis identifies the network and training choices that most strongly influence accuracy.

Fine-tuningMaterial gradientsSensitivity
Transfer pipeline
\[ \theta_s^\star \xrightarrow{\;\text{initialize}\;} \theta_t^{(0)} \xrightarrow{\;\operatorname{fine\!\text{-}tune}(\mathcal{D}_t)\;} \theta_t^\star \]

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 response
\[ \begin{aligned} -\nabla\!\cdot\!\left(K(\boldsymbol{x},\boldsymbol{\xi}) \nabla h_\theta(\boldsymbol{x},\boldsymbol{\xi})\right) &= f(\boldsymbol{x},\boldsymbol{\xi}),\\ \theta^\star &= \underset{\theta}{\operatorname{arg\,min}}\; \mathbb{E}_{\boldsymbol{\xi}}[\mathcal{J}(\theta;\boldsymbol{\xi})]. \end{aligned} \]

Resolve flow responses across heterogeneous material realizations.

05 / Recent collaboration

Neural operators & hybrid solvers

Recent work involving Professor Xiaoying Zhuang and Mohammad Sadegh Eshaghi expands the research from single PDE solutions to reusable operators and hybrid computational systems. VINO learns solution operators from variational physics; PFEM and NOWS use learned fields to initialize trusted numerical solvers; MHNO resolves full interfacial trajectories; and XDEM unifies discrete and phase-field fracture within an enriched energy framework.

VINOPFEMNOWSMHNOXDEM
Learn, initialize, refine
\[ a(\boldsymbol{x}) \xrightarrow{\;\mathcal{G}_\theta\;} \widetilde{u}_0 \xrightarrow{\;\text{numerical refinement}\;} u^\star \]

Connect reusable learned operators with physics and established numerical computation.

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.

Recent neural-operator collaborations

Collaborative work with Mohammad Sadegh Eshaghi extends physics-informed learning to variational operators, finite-element warm starts, interfacial dynamics, transfer learning, and unified fracture mechanics.