Energy approaches to PDEs
Variational energies turn boundary-value problems into optimization objectives that can be solved with neural approximations in computational mechanics.
Read focus ↗Physics-informed computational mechanics
Mechanics from equations.
Learning across domains.
AI4PDE highlights Xiaoying Zhuang's work with collaborators on energy-based machine learning, deep collocation, transfer learning, and multiscale computation for reliable solutions of partial differential equations.
Continuum𝓛(u) = f
Variationalu* = arg min Π(u)
Collocation𝓡(uθ; xᵢ) → 0
Research framework
Physics as the guide.
Learning as the solver.
Research spans variational formulations, physics-informed residuals, material and domain transfer, neural architecture search, and sensitivity analysis.
Equations enforced at sampled points
Physics-informed networks minimize governing-equation and boundary residuals for plates, non-homogeneous media, porous flow, and transient heat transfer.
Read focus ↗Adapt across materials and geometries
Material transfer and domain adaptation reuse learned representations across property gradients, parameters, and complex geometric domains.
Read focus ↗Search architectures for uncertainty
Neural architecture search, global sensitivity analysis, and transfer learning support efficient stochastic simulation of heterogeneous porous media.
Read vision ↗Recent machine learning · 2021–2026
New models.
Broader mechanics.
Recent collaborations by Mohammad Sadegh Eshaghi and Xiaoying Zhuang—with Yizheng Wang and other coauthors—span neural operators, solver acceleration, interfacial dynamics, and PDE learning.
Pretrain Finite Element Method
Physics-informed neural-operator pretraining provides efficient warm starts for accurate finite-element refinement.
Read paper ↗Variational Physics-Informed Neural Operator
VINO embeds variational energy formulations in operator learning and can train without labelled solution data.
Read paper ↗Multi-Head Neural Operator
MHNO predicts full interfacial-dynamics trajectories in one forward pass using temporal projections and explicit connections.
Read paper ↗Kolmogorov–Arnold-Informed Network
KINN studies Kolmogorov–Arnold representations for forward and inverse PDE problems in computational mechanics.
Read paper ↗Representative contributions
From variational ideas
to three-dimensional systems.
01
An energy approach formulates PDE learning through the mechanics of variational principles
02
Deep collocation analyzes Kirchhoff plates without a conventional mesh-based trial space
03
Material transfer and sensitivity analysis address 3D non-homogeneous potential problems
04
Architecture search and transfer learning accelerate stochastic heterogeneous porous flow
Foundational work · energy-based learning
Let mechanics define the learning objective.
The energy approach connects variational principles with neural approximation, providing a clear conceptual and implementation framework for solving PDEs in computational mechanics and applying the method to representative boundary-value problems.