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Yale Statistics & Data Science
Langchen Liu
I am a PhD student in Statistics & Data Science at Yale University. My research is in scientific machine learning, with a current focus on PDE discovery, foundation models for science, and agentic systems for scientific reasoning.
- CurrentFoundPDE, a generative foundation model for symbolic PDE discovery.
- NextPDEScientist, an evaluator-guided system for iterative scientific discovery.
- WritingResearch notes on scientific machine learning, optimization, and paper reading.
Studio
Small interactive sketches for a personal research site: quiet, cursor-aware, and tied to the work rather than decoration alone.
Interactive research desk
Moving details that feel like a working notebook.
This section tests a few directions at once: a portrait that reacts to light, a research constellation, a paper stack, tiny simulations, a Markowitz-style frontier, and a paper companion whose eyes follow the pointer.
Research constellation
hover to pull nearby ideas
Equation ink
terms settle on scroll
E(R_p)=w^T mu
V(R_p)=w^T Sigma w
L(u, grad u, grad^2 u)=0
f* = arg min_f E(f)
Paper stack
hover to lift a page
Publications
papers, preprints, and research documents
Projects
systems, notebooks, experiments
Blog
learning logs and derivations
Cursor-following frontier
risk and return move together
risk 0.42
return 0.68
Research
My current work is organized around symbolic PDE discovery with foundation models and agentic scientific-discovery systems.
Symbolic PDE Discovery
Recovering interpretable governing equations from numerical solution fields, especially when data are sparse or noisy.
PDEs
symbolic structure
spatiotemporal data
Foundation Models for Science
Training reusable models across PDE families so symbolic discovery becomes a learned data-to-sequence problem.
FoundPDE
transformers
pretraining
Agentic Scientific Discovery
Designing systems that propose equations, call evaluators, fit coefficients, inspect feedback, and revise hypotheses.
PDEScientist
tool use
validation loops
Projects
Current projects form a path from one-shot symbolic prediction toward iterative scientific-discovery systems.
FoundPDE
A generative foundation model for data-driven symbolic discovery of one-dimensional time-dependent PDEs. The model maps discretized solution trajectories to symbolic PDE expressions, then refines coefficients with BFGS.
data-to-symbol
Fourier features
coefficient refinement
PDEScientist
An agentic PDE-discovery system where a model proposes candidate equations, analyzes data, calls evaluator tools, fits coefficients, and refines symbolic hypotheses over multiple turns.
agentic AI
PDE evaluator
hypothesis refinement
PDE Evaluation Loops
Benchmark and evaluator design for PDE discovery from raw solution data, including evaluator-owned targets, controlled feature namespaces, residual scoring, coefficient fitting, and trace-based failure analysis.
benchmarks
residual NMSE
rollout checks
Publications
Selected publications, manuscripts, and research documents. The newest work is centered on symbolic PDE discovery.
FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations
Current manuscript on transformer-based data-to-symbol PDE discovery from spatiotemporal solution data, with coefficient refinement, sparse-data evaluation, few-shot adaptation, and step-by-step compositional discovery.
Symbolic PDE Discovery with Foundation Models and Agentic Scientific Discovery
Dissertation prospectus framing FoundPDE as the current foundation and PDEScientist as the next stage toward tool-using, evaluator-grounded scientific-discovery systems.
Federated Scientific Machine Learning for Approximating Functions and Solving Differential Equations With Data Heterogeneity
Journal article on federated scientific machine learning for decentralized function approximation, physics-informed neural networks, and operator learning under data heterogeneity.
Multilingual Language Models are not Multicultural: A Case Study in Emotion
Earlier work studying whether multilingual language models capture culturally specific emotional expression across languages.
Blog
Research notes, paper-reading notes, and technical explanations. Posts live as simple static pages in the repository.
Reading Markowitz 1952 Through Geometry
A post adapted from the Celestial reading note and notebook, with figures and an interactive frontier widget.
How to add a post
Write a Markdown file in the repository's _posts folder. GitHub Pages turns it into a webpage automatically.
Path
A compact academic path from applied mathematics into statistics, machine learning, and scientific AI.
2023 - present
PhD student, Statistics & Data Science
Yale University, New Haven. Advisor: Lu Lu.
2022 - 2023
PhD student, Applied Mathematics and Computational Science
University of Pennsylvania, Philadelphia. Advisor: Lu Lu.
2018 - 2022
BSc, Applied Mathematics
Duke University and Duke Kunshan University.
Contact
The fastest path is email. I also keep professional and code links current on public profiles.