Nicholas M. Boffi

nboffi at andrew dot cmu dot edu

I am an assistant professor in the Machine Learning Department and the Department of Mathematical Sciences at CMU. Previously, I was a Courant Instructor at the Courant Institute of Mathematical Sciences working with Eric Vanden-Eijnden. I completed my PhD in applied mathematics at Harvard co-advised by Jean-Jacques Slotine and Chris Rycroft. A longer bio is here.

Research / Publications / Code / Teaching / Application Materials

Research Group

The goal of our group is to develop a mathematical theory of generative artificial intelligence and to demonstrate that the resulting techniques can solve scientific problems previously thought intractable. We design simple and elegant algorithmic frameworks that elucidate why existing approaches work, reveal principled new methods, and scale to achieve state-of-the-art performance in practice.

Questions we are currently studying include:

Foundations of generative modeling: Why do flow and diffusion models work so well, and how do we move beyond the current paradigm?

In this direction, we have developed:

Stochastic interpolants, a mathematical framework that unifies and underlies modern flow and diffusion models
Flow maps, a theory of accelerated generative models that explains and extends consistency models

Current topics of interest include controlled generation, fine-tuning, inference-time guidance, accelerated sampling, and discrete generation.

Generative models for science and engineering: How can we solve scientific problems that are fundamentally impossible with traditional computational techniques?

We pursue applications in physics, chemistry, biology, vision, and robotics. In each case, we collaborate closely with domain experts to ensure we are solving real problems, as well as targeting challenges that could not be addressed with known domain-specific tools. In this direction, we have developed:

Methods for entropy production estimation in non-equilibrium systems that scale to tens of thousands of dimensions
A critical analysis of generative policies in robotic control, showing they work for different reasons than commonly understood
Methods for accelerated generative modeling over geometric data, combining flow maps with Riemannian geometry for proteins and molecules

Current topics of interest include 3D shape generation, molecular sampling, scientific inverse problems, and rare event estimation in complex dynamical systems.

Beyond our core methodological work, we are early adopters of AI tools and are excited about a future of scientific research deeply integrated with AI. We investigate new ways to leverage language models and agentic automation to dramatically accelerate the pace of research within our group and beyond.

PhD Students

Jerry Huang
Computer Science Department

Stephen Huan
Computer Science Department
co-advised with Andrej Risteski

Masters Students

Kartik Nair
Machine Learning Department

Sheel Shah
Machine Learning Department

Undergraduate Students

Ishin Shah
Department of Mathematical Sciences
co-advised with Max Simchowitz

Publications

Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds

Oscar Davis, Michael S. Albergo, Nicholas M. Boffi, Michael M. Bronstein, Avishek Joey Bose

Preprint, 2025

arXiv

Sublinear iterations can suffice even for DDPMs

Matthew S. Zhang, Stephen Huan, Jerry Huang, Nicholas M. Boffi, Sitan Chen, Sinho Chewi

Preprint, 2025

arXiv

Eigenfunction Extraction for Ordered Representation Learning

Burak Varıcı, Che-Ping Tsai, Ritabrata Ray, Nicholas M. Boffi, Pradeep Ravikumar

Preprint, 2025

arXiv

How to build a consistency model: Learning flow maps via self-distillation

Nicholas M. Boffi, Michael S. Albergo, Eric Vanden-Eijnden

Neural Information Processing Systems, 2025

arXiv / code

BoltzNCE: Learning Likelihoods for Boltzmann Generation with Stochastic Interpolants and Noise Contrastive Estimation

Rishal Aggarwal, Jacky Chen, Nicholas M. Boffi, David Ryan Koes

Neural Information Processing Systems, 2025

arXiv

Model-free learning of probability flows: Elucidating the nonequilibrium dynamics of flocking

Nicholas M. Boffi, Eric Vanden-Eijnden

Physical Review Letters, 2025

arXiv / code

Shallow diffusion networks provably learn hidden low-dimensional structure

Nicholas M. Boffi, Arthur Jacot, Stephen Tu, Ingvar M. Ziemann

International Conference on Learning Representations, 2025

arXiv / code

Flow map matching with stochastic interpolants: A mathematical framework for consistency models

Nicholas M. Boffi*, Michael S. Albergo*, Eric Vanden-Eijnden

Transactions on Machine Learning Research, 2025

arXiv / code

Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

Michael S. Albergo*, Nicholas M. Boffi*, Eric Vanden-Eijnden

Journal of Machine Learning Research, 2025

First appeared on arXiv 2023

arXiv / code 1 / code 2

Test-time scaling of diffusions with flow maps

Amirmojtaba Sabour, Michael S. Albergo, Carles Domingo-Enrich, Nicholas M. Boffi, Sanja Fidler, Karsten Kreis, Eric Vanden-Eijnden

Preprint, 2025

arXiv

Much Ado About Noising: Dispelling the Myths of Generative Robotic Control

Chaoyi Pan, Giri Anantharaman, Nai-Chieh Huang, Claire Jin, Daniel Pfrommer, Chenyang Yuan, Frank Permenter, Guannan Qu, Nicholas M. Boffi†, Guanya Shi†, Max Simchowitz†

Preprint, 2025

arXiv

Pre-Generating Multi-Difficulty PDE Data for Few-Shot Neural PDE Solvers

Naman Choudhary, Vedant Singh, Ameet Talwalkar, Nicholas M. Boffi, Mikhail Khodak, Tanya Marwah

Preprint, 2025

arXiv

Joint Distillation for Fast Likelihood Evaluation and Sampling in Flow-based Models

Xinyue Ai, Yutong He, Albert Gu, Ruslan Salakhutdinov, J. Zico Kolter, Nicholas M. Boffi, Max Simchowitz

Preprint, 2025

arXiv

Multimarginal generative modeling with stochastic interpolants

Michael S. Albergo, Nicholas M. Boffi, Michael Lindsey, Eric Vanden-Eijnden

International Conference on Learning Representations, 2024

arXiv

Deep learning probability flows and entropy production rates in active matter

Nicholas M. Boffi, Eric Vanden-Eijnden

Proceedings of the National Academy of Sciences, 2024

arXiv / paper / code

Stochastic interpolants with data-dependent couplings

Michael S. Albergo, Mark Goldstein, Nicholas M. Boffi, Rajesh Ranganath, Eric Vanden-Eijnden

International Conference on Machine Learning, 2024 (Spotlight)

arXiv

Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden

International Conference on Machine Learning, 2024

arXiv / code

SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers

Nanye Ma, Mark Goldstein, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden, Saining Xie

European Conference on Computer Vision, 2024

arXiv / paper

Agile Catching with Whole-Body MPC and Blackbox Policy Learning

Saminda Abeyruwan, Alex Bewley, Nicholas M. Boffi, Krzysztof Marcin Choromanski, David B D'Ambrosio, Deepali Jain, Pannag R Sanketi, Anish Shankar, Vikas Sindhwani, Sumeet Singh, Jean-Jacques Slotine, Stephen Tu

Learning for Dynamics and Control, 2023

paper

Thumbnail for How microscopic epistasis and clonal interference shape the fitness trajectory in a spin glass model of microbial long-term evolution

How microscopic epistasis and clonal interference shape the fitness trajectory in a spin glass model of microbial long-term evolution

Nicholas M. Boffi, Yipei Guo, Chris H. Rycroft, Ariel Amir

eLife, 2023

paper / code / bioRxiv

Probability flow solution of the Fokker-Planck equation

Nicholas M. Boffi, Eric Vanden-Eijnden

Machine Learning Science and Technology, 2023

arXiv / paper / code

The role of optimization geometry in single neuron learning

Nicholas M. Boffi*, Stephen Tu*, Jean-Jacques E. Slotine

International Conference on Artificial Intelligence and Statistics, 2022

arXiv / paper

Adversarially robust stability certificates can be sample efficient

Thomas Zhang, Stephen Tu, Nicholas M. Boffi, Jean-Jacques Slotine, Nikolai Matni

Learning for Dynamics and Control, 2022

arXiv / paper

Non-parametric adaptive control and prediction: theory and randomized algorithms

Nicholas M. Boffi*, Stephen Tu*, Jean-Jacques Slotine

Journal of Machine Learning Research (conference version at CDC 2021), 2022

arXiv / paper

Implicit regularization and momentum algorithms in nonlinearly parameterized adaptive control and prediction

Nicholas M. Boffi, Jean-Jacques E. Slotine

Neural Computation, 2021 (Cover)

arXiv / paper

Regret bounds for adaptive nonlinear control

Nicholas M. Boffi*, Stephen Tu*, Jean-Jacques E. Slotine

Learning for Dynamics and Control, 2021 (Oral)

arXiv / paper

Manifold learning for coarse-graining atomistic simulations: Application to amorphous solids

Katiana Kontolati, Darius Alix-Williams, Nicholas M. Boffi, Michael L. Falk, Chris H. Rycroft, Michael D. Shields

Acta Materialia, 2021

arXiv / paper

A continuous-time analysis of distributed stochastic gradient

Nicholas M. Boffi, Jean-Jacques E. Slotine

Neural Computation, 2020

arXiv / paper

Parallel three-dimensional simulations of quasi-static elastoplastic solids

Nicholas M. Boffi, Chris H. Rycroft

Computer Physics Communications, 2020

arXiv / paper / code

Coordinate transformation methodology for simulating quasi-static elastoplastic solids

Nicholas M. Boffi, Chris H. Rycroft

Physical Review E, 2020

arXiv / paper / code

Learning stability certificates from data

Nicholas M. Boffi*, Stephen Tu*, Nikolai Matni, Jean-Jacques E. Slotine, Vikas Sindhwani

Conference on Robot Learning, 2020

arXiv / paper

Asymptotic behavior and interpretation of virtual states: the effects of confinement and of basis sets

Nicholas M. Boffi, Manish Jain, Amir Natan

Journal of Chemical Physics, 2016

paper / code

Efficient computation of the Hartree-Fock Exchange in real-space with projection operators

Nicholas M. Boffi, Manish Jain, Amir Natan

Journal of Chemical Theory and Computation, 2016

paper / code

Characterizing the inverses of block tridiagonal, block Toeplitz matrices

Nicholas M. Boffi, Judith C. Hill, Matthew G. Reuter

Computational Science and Discovery, 2014

paper

The role of dimensionality in the decay of surface effects

Matthew G. Reuter, Nicholas M. Boffi, Mark A. Ratner, Tamar Seideman

Journal of Chemical Physics, 2013

paper

Code

We are strong advocates for reproducible science and open-source software. We believe that they have been key drivers of the recent advances in AI research, and we make all our code publicly available to continue this tradition.

jax-interpolants
A JAX implementation of the stochastic interpolant framework for generative modeling
jax-edm2
A JAX implementation of NVIDIA's EDM2 U-Net architecture
flow-maps (with Michael Albergo and Eric Vanden Eijnden)
A JAX implementation of the self-distillation framework for learning flow maps and consistency models
model-free probability flows (with Eric Vanden Eijnden)
Code for model-free learning of probability flows in nonequilibrium dynamics
structured diffusion (with Arthur Jacot and Stephen Tu)
Experiments verifying provable guarantees for when shallow diffusion networks can learn hidden structure
flow map matching (with Michael Albergo and Eric Vanden Eijnden)
A JAX implementation of the flow map matching framework for learning flow maps and consistency models
forecasting with interpolants (with Mengjian Hua, Yifan Chen, Mark Goldstein, Eric Vanden Eijnden, and Michael Albergo -- mostly written by the first three authors!)
An implementation of probabilistic forecasting with stochastic interpolants
active probability flows (with Eric Vanden Eijnden)
Code for learning probability flows and entropy production rates in active matter
stochastic interpolants (with Michael Albergo and Eric Vanden Eijnden)
An implementation of the unifying stochastic interpolant framework for flows and diffusions in generative modeling
score-based transport modeling (with Eric Vanden Eijnden)
Code for constructing a probability flow solution of the Fokker-Planck equation
spin glass evolutionary dynamics (with Yipei Guo, Chris Rycroft, and Ariel Amir)
High-resolution C++ simulation code to model Lenski's Long-Term Evolution Experiment, taking into account microscopic epistasis and clonal interference in microbial evolution
shear transformation zone++ (with Chris Rycroft)
Three-dimensional MPI-based C++ code for parallel simulation of quasi-static deformation in elastoplastic solids
real-space Hartree-Fock (with Amir Natan, note: only the Hartree-Fock implementation)
Parallelized Fortran code to efficiently compute the Hartree-Fock exchange via the use of projection operators onto occupied and virtual states

Teaching

I teach mathematically- and computationally-oriented courses in applied mathematics and machine learning. Below is a list of past, present, and future offerings.

Carnegie Mellon, Spring 2025: Methods of Optimization

Carnegie Mellon, Fall 2024: Introduction to PDEs: A Computational Approach

NYU Courant, Spring 2024: Honors Numerical Analysis

NYU Courant, Fall 2023: Linear and Nonlinear Optimization

NYU Courant, Spring 2023: Linear and Nonlinear Optimization

NYU Courant, Fall 2022: Numerical Analysis

NYU Courant, Spring 2022: Linear and Nonlinear Optimization

NYU Courant, Fall 2021: Numerical Analysis

Northwestern, Full Year 2014: Introduction to Computer Programming for Integrated Science