Energy approaches to PDEs
Variational energies turn boundary-value problems into optimization objectives that can be solved with neural approximations in computational mechanics.
Machine learning for PDEs
Learn the solution.
Build the physics in.
AI4PDE is led by Professor Xiaoying Zhuang's research in machine learning for computational mechanics. The work spans physics-informed networks, deep collocation, transfer learning, neural architecture search, multiscale models, and scientific machine learning.
Representationu ≈ uθ(x)
Learning objectiveθ* = arg min J(θ)
Physics constraint𝓡(uθ; xᵢ) → 0
Machine learning framework
Learn solutions.
Transfer knowledge.
Research moves from neural representations and physics-informed objectives to transfer learning, architecture search, uncertainty analysis, and reusable PDE models.
Equations enforced at sampled points
Physics-informed networks minimize governing-equation and boundary residuals for plates, non-homogeneous media, and porous flow.
Adapt across materials and geometries
Material transfer and fine-tuning reuse learned representations across property gradients, parameters, boundary conditions, and related PDE problems.
Search architectures for uncertainty
Neural architecture search, global sensitivity analysis, and transfer learning support efficient stochastic simulation of heterogeneous porous media.
Latest research · checked 25 Aug 2026
Latest
machine-learning papers.
Ordered by first online publication or DOI registration date. The selection follows Professor Xiaoying Zhuang's newest work in scientific machine learning for mechanics and PDEs.
Towards unified AI-driven fracture mechanics
XDEM unifies discrete and phase-field fracture mechanics through enriched energy-based neural fields.
Read paper ↗ Research context →Deep Energy Method with LLM-assisted geometry modeling
An open-source Streamlit platform connects large-language-model-assisted geometry creation with energy-based neural PDE solution.
Read paper ↗ Research context →Neural Operator Warm Starts
NOWS uses learned solution operators to initialize established iterative solvers, reducing computation while retaining their stability and convergence guarantees.
Read paper ↗ Research context →Highly cited foundations · OpenAlex
The two most-cited
machine-learning papers.
Citation counts are a dated snapshot rather than a permanent ranking. Counts shown here were retrieved from OpenAlex on 25 August 2026.
2,045OpenAlex citations ↗
An energy approach to PDEs via machine learning
A foundational connection between variational mechanics, neural approximation, and physics-based objectives for solving partial differential equations.
Read paper ↗ Visual sample →Deep collocation for Kirchhoff plate bending
A mesh-free neural formulation enforces a fourth-order plate equation and boundary conditions at collocation points.
Read paper ↗ Visual sample →Machine learning in practice
From learned fields
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 machine learning
Train neural fields with physical energy.
The energy approach connects neural approximation with variational principles, providing a clear machine-learning framework for representing PDE solutions, constructing physics-based objectives, and solving boundary-value problems in computational mechanics.