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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Langchen Liu</title>
<meta name="description" content="Langchen Liu is a PhD student in Statistics & Data Science at Yale University working on symbolic PDE discovery, foundation models for science, and agentic scientific-discovery systems.">
<meta property="og:title" content="Langchen Liu">
<meta property="og:description" content="PhD student in Statistics & Data Science at Yale University.">
<meta property="og:type" content="website">
<meta property="og:url" content="https://decoderliu.github.io/">
<meta property="og:image" content="https://decoderliu.github.io/assets/langchen-liu.jpg">
<link rel="stylesheet" href="assets/interactive-studio.css">
<style>
:root {
color-scheme: light;
--ink: #14171f;
--muted: #59616f;
--quiet: #eef0f4;
--paper: #fbfaf7;
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--shadow: 0 24px 70px rgba(22, 30, 45, 0.15);
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* {
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html {
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font-family: Charter, "Iowan Old Style", "Palatino Linotype", Palatino, Georgia, serif;
color: var(--ink);
background: linear-gradient(180deg, #fbfaf7 0%, #f4f6f8 100%);
}
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linear-gradient(rgba(20, 23, 31, 0.045) 1px, transparent 1px),
linear-gradient(90deg, rgba(20, 23, 31, 0.045) 1px, transparent 1px);
background-size: 44px 44px;
mask-image: linear-gradient(to bottom, black, transparent 92%);
}
a {
color: inherit;
text-decoration: none;
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a:hover {
color: var(--blue);
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.shell {
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header {
position: sticky;
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z-index: 10;
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backdrop-filter: blur(18px);
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.nav {
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width: 34px;
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display: flex;
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display: inline-flex;
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}
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border-color: #173c92;
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color: var(--blue);
font-weight: 720;
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position: relative;
justify-self: end;
width: min(100%, 350px);
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display: block;
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object-fit: cover;
object-position: 50% 18%;
filter: saturate(0.9) contrast(1.04);
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.caption {
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section {
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grid-template-columns: 0.42fr 1fr;
gap: 40px;
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margin-bottom: 34px;
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h2 {
margin: 0;
color: var(--muted);
font-size: 0.92rem;
letter-spacing: 0.16em;
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.section-head p {
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box-shadow: 0 10px 34px rgba(22, 30, 45, 0.07);
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.card-link,
.publication-link {
transition: border-color 160ms ease, transform 160ms ease, box-shadow 160ms ease;
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.card-link:hover,
.publication-link:hover {
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color: inherit;
transform: translateY(-2px);
box-shadow: 0 18px 42px rgba(22, 30, 45, 0.1);
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.card-link:hover h3,
.publication-link:hover h3 {
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.publication .meta,
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font-size: 0.9rem;
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.event p {
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gap: 24px;
padding: 24px;
border: 1px solid var(--line);
border-radius: 8px;
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box-shadow: 0 10px 34px rgba(22, 30, 45, 0.06);
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.publication h3 {
margin: 0 0 8px;
font-size: 1.12rem;
}
.publication p {
margin: 0 0 10px;
color: var(--muted);
}
.publication a {
color: var(--blue);
font-weight: 720;
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.contact-grid {
display: grid;
grid-template-columns: repeat(4, minmax(0, 1fr));
gap: 12px;
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.contact-link {
min-height: 96px;
padding: 18px;
border: 1px solid var(--line);
border-radius: 8px;
background: rgba(255, 255, 255, 0.82);
box-shadow: 0 10px 30px rgba(22, 30, 45, 0.06);
}
.contact-link span {
display: block;
margin-bottom: 8px;
color: var(--muted);
font-size: 0.82rem;
text-transform: uppercase;
letter-spacing: 0.12em;
}
footer {
padding: 34px 0 52px;
color: var(--muted);
font-size: 0.92rem;
}
@media (max-width: 880px) {
.nav {
min-height: 62px;
}
nav {
display: none;
}
.hero,
.section-head,
.event,
.publication,
.contact-grid {
grid-template-columns: 1fr;
}
.hero {
gap: 36px;
padding-top: 42px;
}
.focus-list li {
grid-template-columns: 1fr;
gap: 2px;
}
.portrait-panel {
justify-self: start;
width: min(100%, 340px);
}
.status-card {
position: static;
width: auto;
margin-top: 12px;
}
.portrait-card {
transform: none;
}
.grid {
grid-template-columns: 1fr;
}
}
</style>
</head>
<body>
<header>
<div class="shell nav">
<a class="mark" href="#top" aria-label="Langchen Liu home">
<span>LL</span>
<span>Langchen Liu</span>
</a>
<nav aria-label="Primary navigation">
<a href="#studio">Studio</a>
<a href="#research">Research</a>
<a href="#projects">Projects</a>
<a href="#publications">Publications</a>
<a href="#blog">Blog</a>
<a href="#path">Path</a>
<a href="#contact">Contact</a>
</nav>
</div>
</header>
<main id="top">
<div class="shell hero">
<div>
<p class="eyebrow">Yale Statistics & Data Science</p>
<h1>Langchen Liu</h1>
<p class="lede">
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.
</p>
<ul class="focus-list" aria-label="Current focus areas">
<li><strong>Current</strong><span>FoundPDE, a generative foundation model for symbolic PDE discovery.</span></li>
<li><strong>Next</strong><span>PDEScientist, an evaluator-guided system for iterative scientific discovery.</span></li>
<li><strong>Writing</strong><span>Research notes on scientific machine learning, optimization, and paper reading.</span></li>
</ul>
<div class="actions" aria-label="Primary links">
<a class="button primary" href="mailto:langchen.liu@yale.edu">Email</a>
<a class="button" href="https://scholar.google.com/citations?hl=en&user=CQZcPZQAAAAJ">Scholar</a>
<a class="button" href="https://www.linkedin.com/in/langchen-liu/">LinkedIn</a>
<a class="button" href="https://github.com/DecoderLiu">GitHub</a>
</div>
</div>
<aside class="portrait-panel" aria-label="Profile summary">
<div class="portrait-card">
<img src="assets/langchen-liu.jpg" alt="Langchen Liu">
<div class="caption">
<span>DecoderLiu</span>
<span>New Haven</span>
</div>
</div>
<div class="status-card">
<strong>Research focus</strong>
<span>Scientific machine learning, PDE discovery, and AI systems for scientific reasoning.</span>
</div>
</aside>
</div>
<section id="studio" data-studio>
<div class="shell">
<div class="section-head">
<h2>Studio</h2>
<p>Small interactive sketches for a personal research site: quiet, cursor-aware, and tied to the work rather than decoration alone.</p>
</div>
<div class="studio-grid">
<div class="studio-intro studio-reveal">
<p class="studio-kicker">Interactive research desk</p>
<h2>Moving details that feel like a working notebook.</h2>
<p>
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.
</p>
</div>
<div class="studio-portrait studio-reveal" data-cursor-card>
<img src="assets/langchen-liu.jpg" alt="Langchen Liu">
<span class="studio-portrait__ring" aria-hidden="true"></span>
</div>
<div class="studio-companion studio-reveal" data-cursor-card aria-label="Cursor-following paper companion">
<svg viewBox="0 0 240 180" role="img" aria-labelledby="paper-companion-title">
<title id="paper-companion-title">A paper-like companion whose eyes follow the cursor</title>
<path class="studio-paper-shape" d="M44 38h112l40 40v76H44z" />
<path class="studio-fold" d="M156 38v40h40" />
<path class="studio-ear" d="M58 39 78 16l19 22" />
<path class="studio-ear" d="M128 38 151 15l14 25" />
<circle cx="93" cy="90" r="18" class="studio-eye" />
<circle cx="143" cy="90" r="18" class="studio-eye" />
<circle cx="93" cy="90" r="6" class="studio-pupil" data-eye />
<circle cx="143" cy="90" r="6" class="studio-pupil" data-eye />
<path class="studio-smile" d="M107 120c10 9 24 9 34 0" />
<path class="studio-line" d="M70 137h102" />
</svg>
</div>
<div class="studio-constellation studio-reveal" data-cursor-card>
<div class="studio-panel-title">
<span>Research constellation</span>
<small>hover to pull nearby ideas</small>
</div>
<svg viewBox="0 0 520 300" data-constellation aria-label="Interactive research topic constellation">
<line x1="102" y1="88" x2="243" y2="132" />
<line x1="243" y1="132" x2="394" y2="76" />
<line x1="243" y1="132" x2="338" y2="220" />
<line x1="162" y1="222" x2="243" y2="132" />
<line x1="162" y1="222" x2="338" y2="220" />
<g data-node data-x="102" data-y="88"><circle r="14" /><text>PDE</text></g>
<g data-node data-x="243" data-y="132"><circle r="18" /><text>Scientific ML</text></g>
<g data-node data-x="394" data-y="76"><circle r="14" /><text>Symbolic</text></g>
<g data-node data-x="162" data-y="222"><circle r="14" /><text>Quant</text></g>
<g data-node data-x="338" data-y="220"><circle r="14" /><text>Agents</text></g>
</svg>
</div>
<div class="studio-equations studio-reveal" data-cursor-card>
<div class="studio-panel-title">
<span>Equation ink</span>
<small>terms settle on scroll</small>
</div>
<p><span>E(R_p)=w^T mu</span></p>
<p><span>V(R_p)=w^T Sigma w</span></p>
<p><span>L(u, grad u, grad^2 u)=0</span></p>
<p><span>f* = arg min_f E(f)</span></p>
</div>
<div class="studio-stack studio-reveal" data-cursor-card>
<div class="studio-panel-title">
<span>Paper stack</span>
<small>hover to lift a page</small>
</div>
<a class="studio-paper" href="#publications">
<strong>Publications</strong>
<span>papers, preprints, and research documents</span>
</a>
<a class="studio-paper" href="#projects">
<strong>Projects</strong>
<span>systems, notebooks, experiments</span>
</a>
<a class="studio-paper" href="#blog">
<strong>Blog</strong>
<span>learning logs and derivations</span>
</a>
</div>
<div class="studio-frontier studio-reveal" data-cursor-card>
<div class="studio-panel-title">
<span>Cursor-following frontier</span>
<small>risk and return move together</small>
</div>
<canvas width="520" height="260" data-frontier></canvas>
<div class="studio-frontier__readout">
<span>risk <b data-risk>0.42</b></span>
<span>return <b data-return>0.68</b></span>
</div>
</div>
<div class="studio-bench studio-reveal">
<div class="studio-panel-title">
<span>Lab bench</span>
<small>micro-simulations by route</small>
</div>
<a class="studio-tool" href="#blog">
<canvas width="96" height="58" data-spark="wave"></canvas>
<span>Notes</span>
</a>
<a class="studio-tool" href="#projects">
<canvas width="96" height="58" data-spark="tree"></canvas>
<span>Projects</span>
</a>
<a class="studio-tool" href="#publications">
<canvas width="96" height="58" data-spark="curve"></canvas>
<span>Papers</span>
</a>
</div>
</div>
</div>
</section>
<section id="research">
<div class="shell">
<div class="section-head">
<h2>Research</h2>
<p>My current work is organized around symbolic PDE discovery with foundation models and agentic scientific-discovery systems.</p>
</div>
<div class="grid">
<a class="card card-link" href="research/symbolic-pde-discovery/">
<span class="meta">Problem</span>
<h3>Symbolic PDE Discovery</h3>
<p>Recovering interpretable governing equations from numerical solution fields, especially when data are sparse or noisy.</p>
<div class="tag-row">
<span class="tag">PDEs</span>
<span class="tag">symbolic structure</span>
<span class="tag">spatiotemporal data</span>
</div>
</a>
<a class="card card-link" href="research/foundation-models-for-science/">
<span class="meta">Method</span>
<h3>Foundation Models for Science</h3>
<p>Training reusable models across PDE families so symbolic discovery becomes a learned data-to-sequence problem.</p>
<div class="tag-row">
<span class="tag">FoundPDE</span>
<span class="tag">transformers</span>
<span class="tag">pretraining</span>
</div>
</a>
<a class="card card-link" href="research/agentic-scientific-discovery/">
<span class="meta">Next stage</span>
<h3>Agentic Scientific Discovery</h3>
<p>Designing systems that propose equations, call evaluators, fit coefficients, inspect feedback, and revise hypotheses.</p>
<div class="tag-row">
<span class="tag">PDEScientist</span>
<span class="tag">tool use</span>
<span class="tag">validation loops</span>
</div>
</a>
</div>
</div>
</section>
<section id="projects">
<div class="shell">
<div class="section-head">
<h2>Projects</h2>
<p>Current projects form a path from one-shot symbolic prediction toward iterative scientific-discovery systems.</p>
</div>
<div class="grid">
<a class="card card-link" href="projects/foundpde/">
<span class="meta">Current manuscript</span>
<h3>FoundPDE</h3>
<p>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.</p>
<div class="tag-row">
<span class="tag">data-to-symbol</span>
<span class="tag">Fourier features</span>
<span class="tag">coefficient refinement</span>
</div>
</a>
<a class="card card-link" href="projects/pdescientist/">
<span class="meta">Dissertation direction</span>
<h3>PDEScientist</h3>
<p>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.</p>
<div class="tag-row">
<span class="tag">agentic AI</span>
<span class="tag">PDE evaluator</span>
<span class="tag">hypothesis refinement</span>
</div>
</a>
<a class="card card-link" href="projects/pde-evaluation-loops/">
<span class="meta">Research infrastructure</span>
<h3>PDE Evaluation Loops</h3>
<p>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.</p>
<div class="tag-row">
<span class="tag">benchmarks</span>
<span class="tag">residual NMSE</span>
<span class="tag">rollout checks</span>
</div>
</a>
</div>
</div>
</section>
<section id="publications">
<div class="shell">
<div class="section-head">
<h2>Publications</h2>
<p>Selected publications, manuscripts, and research documents. The newest work is centered on symbolic PDE discovery.</p>
</div>
<div class="publication-list">
<a class="publication publication-link" href="publications/foundpde/">
<div>
<span class="meta">Manuscript</span>
</div>
<div>
<h3>FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations</h3>
<p>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.</p>
</div>
</a>
<a class="publication publication-link" href="publications/prospectus/">
<div>
<span class="meta">Prospectus</span>
</div>
<div>
<h3>Symbolic PDE Discovery with Foundation Models and Agentic Scientific Discovery</h3>
<p>Dissertation prospectus framing FoundPDE as the current foundation and PDEScientist as the next stage toward tool-using, evaluator-grounded scientific-discovery systems.</p>
</div>
</a>
<a class="publication publication-link" href="publications/federated-sciml/">
<div>
<span class="meta">IEEE TNNLS 2025</span>
</div>
<div>
<h3>Federated Scientific Machine Learning for Approximating Functions and Solving Differential Equations With Data Heterogeneity</h3>
<p>Journal article on federated scientific machine learning for decentralized function approximation, physics-informed neural networks, and operator learning under data heterogeneity.</p>
</div>
</a>
<a class="publication publication-link" href="publications/multilingual-emotion/">
<div>
<span class="meta">WASSA 2023</span>
</div>
<div>
<h3>Multilingual Language Models are not Multicultural: A Case Study in Emotion</h3>
<p>Earlier work studying whether multilingual language models capture culturally specific emotional expression across languages.</p>
</div>
</a>
</div>
</div>
</section>
<section id="blog">
<div class="shell">
<div class="section-head">
<h2>Blog</h2>
<p>Research notes, paper-reading notes, and technical explanations. Posts live as simple static pages in the repository.</p>
</div>
<div class="grid">
<article class="card">
<span class="meta">Blog index</span>
<h3><a class="text-link" href="blog/">All posts</a></h3>
<p>The blog landing page lists every post. This is the page to update whenever a new post is added.</p>
</article>
<article class="card">
<span class="meta">Reading note</span>
<h3><a class="text-link" href="blog/markowitz-portfolio-selection/">Reading Markowitz 1952 Through Geometry</a></h3>
<p>A post adapted from the Celestial reading note and notebook, with figures and an interactive frontier widget.</p>
</article>
<article class="card">
<span class="meta">Writing guide</span>
<h3><a class="text-link" href="blog/how-to-write/">How to add a post</a></h3>
<p>Write a Markdown file in the repository's <code>_posts</code> folder. GitHub Pages turns it into a webpage automatically.</p>
</article>
</div>
</div>
</section>
<section id="path">
<div class="shell">
<div class="section-head">
<h2>Path</h2>
<p>A compact academic path from applied mathematics into statistics, machine learning, and scientific AI.</p>
</div>
<div class="timeline">
<article class="event">
<div class="date">2023 - present</div>
<div>
<h3>PhD student, Statistics & Data Science</h3>
<p>Yale University, New Haven. Advisor: Lu Lu.</p>
</div>
</article>
<article class="event">
<div class="date">2022 - 2023</div>
<div>
<h3>PhD student, Applied Mathematics and Computational Science</h3>
<p>University of Pennsylvania, Philadelphia. Advisor: Lu Lu.</p>
</div>
</article>
<article class="event">
<div class="date">2018 - 2022</div>
<div>
<h3>BSc, Applied Mathematics</h3>
<p>Duke University and Duke Kunshan University.</p>
</div>
</article>
</div>
</div>
</section>
<section id="contact">
<div class="shell">
<div class="section-head">
<h2>Contact</h2>
<p>The fastest path is email. I also keep professional and code links current on public profiles.</p>
</div>
<div class="contact-grid">
<a class="contact-link" href="mailto:langchen.liu@yale.edu">
<span>Email</span>
langchen.liu@yale.edu
</a>
<a class="contact-link" href="https://github.com/DecoderLiu">
<span>Code</span>
GitHub
</a>
<a class="contact-link" href="https://scholar.google.com/citations?hl=en&user=CQZcPZQAAAAJ">
<span>Publications</span>
Google Scholar
</a>
<a class="contact-link" href="https://www.linkedin.com/in/langchen-liu/">
<span>Social</span>
LinkedIn
</a>
</div>
</div>
</section>
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