Scientific Computing and Learning

Yiping Lu

I am a tenure-track assistant professor at Beijing International Center for Mathematical Research, Peking University.

I develop mathematical foundations and scalable algorithms at the intersection of machine learning, numerical computation, applied probability, and scientific discovery.

Diffusion Models
Generative Model Inference
Neural Network Optimizers
Physical AI
Computational Mathematics

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yipinglu@bicmr.pku.edu.cn

Office: No.78 Jingchunyuan 78105W-1

Current position Assistant Professor · BICMR · PKU

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Work with us

I am actively recruiting undergraduate students, graduate students, and postdocs to join my research group. Interested candidates are encouraged to email yipinglu [at] bicmr.pku.edu.cn.

Students and opportunities

Research Scaling laws for learning and inference

Methods Machine Learning Theory · Applied Probability · Computational Math

Education Stanford PhD · PKU BS

Lab SCALE Lab Scientific Computing And LEarning

Research directions

How can learning systems scale reliably?

Inference Time Scaling

Monte Carlo, control, and sequential methods that convert additional inference computation into reliable accuracy gains.

Selected works (2)

Generative AI · 2026

URGE

Unbiased derivative free inference time scaling for diffusion models through sequential Monte Carlo on path measures.

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LLM Reasoning Theory · 2026

On the Power of Approximate Reward Models for Inference Time Scaling

A theory of when approximate reward models reduce the complexity of long-horizon LLM reasoning from exponential to polynomial through SMC inference-time scaling.

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Scientific Machine Learning

Structure preserving learning for PDEs, operator learning, uncertainty quantification, and simulation calibrated correction.

Selected works (5)

AI for Science · ICLR 2026

Simulation Calibrated Scientific ML

Inference time defect correction improves high dimensional PDE solvers without retraining the learned model.

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Equation Discovery · ICML 2018

PDE Net: Learning PDEs from Data

Learns differential operators and nonlinear dynamics through constrained convolution filters to identify PDEs from observed data.

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Equation Discovery · JCP 2019

PDE Net 2.0

Combines learnable differential operators with symbolic neural networks to recover explicit PDE models and predict their dynamics.

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PDE Learning Theory · ICLR 2022

Machine Learning for Elliptic PDEs

Establishes sharp generalization bounds and minimax rates for PINNs and a modified Deep Ritz method in a prototype elliptic PDE setting.

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Operator Learning · ICLR 2023 Spotlight

Minimax Optimal Kernel Operator Learning via Multilevel Training

Develops minimax optimal rates and a multilevel algorithm for learning linear operators between infinite dimensional function spaces.

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Optimization and Reliability

Width and depth stable optimization geometry, predictable hyperparameter transfer, and robust learning algorithms.

Selected works (3)

Optimization · 2026

Scaling Neural Optimizers

Matrix operator norm geometry explains width scaling, row and column normalization, and hyperparameter transfer.

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Statistics · 2026

Fragility of Interpolators

Heavy tailed risk and high dimensional large deviations reveal failure modes hidden by benign average case behavior.

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Feature Geometry · ICLR 2022

An Unconstrained Layer Peeled Perspective on Neural Collapse

Studies neural collapse through the implicit bias of gradient flow and the optimization geometry of an unconstrained model of features and classifiers.

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Agentic Mathematical Reasoning

Representations and search procedures that help AI systems discover, verify, and communicate mathematical structure.

Selected works (1)

Probability · 2026

Signed BAR Conjecture

Uniqueness in the Harrison–Reiman class and a completely S class obstruction for a longstanding problem in reflected Brownian motion.

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Differential Equations for ML

Numerical differential equations, optimal control, and mean field limits provide principles for neural network architecture, efficient training, and optimization theory.

Selected works (3)

Network Architectures · ICML 2018

Beyond Finite Layer Neural Networks

Interprets neural architectures as numerical discretizations of differential equations and uses linear multistep methods to design more efficient residual networks.

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Optimal Control · NeurIPS 2019

You Only Propagate Once (YOPO)

Formulates adversarial training as a differential game and uses Pontryagin’s maximum principle to reduce repeated propagation through the full network.

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Optimization Theory · ICML 2020

A Mean Field Analysis of Deep ResNet and Beyond

Develops a continuum model of deep residual networks and establishes optimization guarantees through mean field analysis and overparameterization from depth.

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Browse selected publications

SCALE Lab · Research notes

MAGI Blog

Understanding generative AI through mathematics.

All posts

2026.10.10 Probabilistic Modeling

Let Diffusion Models Think Longer: From URGE to Stein Guidance without Gradients

From shared ancestry to independent rejection, from imperfect values to path correction: finite differences and Stein’s identity explain guidance using only function values.

Diffusion modelsInference time computationURGE

Teaching archives

Course materials

Northwestern · IEMS 407 Uncertainty Quantification Open course archive → Northwestern · IEMS 304 Statistical Learning for Data Analysis Open course archive → Northwestern · IEMS 402 Statistical Learning Open course archive → NYU Courant · MATH-UA 140 Linear Algebra Open course archive →

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