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Computational and Mathematical Engineering
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  5. Doctor of Philosophy

Doctor of Philosophy

PhD
  • Institute for Computational and Mathematical Engr opens in a new tab
Program Minimum Units
135 Units
University Minimum Units
135 Units
Program Length
6 Years
Interests
Data and Information Sciences, Technology
Academic Areas
Engineering & Computer Science, Mathematics & Data Sciences

Overview

ICME is a degree-granting (MS/PhD) interdisciplinary institute at the intersection of mathematics, computing, engineering, and applied sciences. ICME was founded in 2004, building upon the Scientific Computing and Computational Mathematics Program (est. 1989).

At ICME, we design state-of-the-art mathematical and computational models, methods, and algorithms for engineering and science applications. The program collaborates closely with engineers and scientists in academia and industry to develop improved computational approaches and advance disciplinary fields. In particular, it leverages Stanford’s strength in engineering applications in the physical, biological, mathematical, and information sciences and has established connections with nearly 20 departments across five schools at Stanford.

The program identifies research areas that would benefit from a multidisciplinary approach in which computational mathematics is critical. This multidisciplinary intellectual environment is a core strength of ICME, with interaction among students and faculty with diverse backgrounds and expertise. Students and faculty are active in many research areas: aerodynamics and space applications, fluid dynamics, protein folding, data science, including machine learning and recommender systems, ocean dynamics, climate modeling, reservoir engineering, computer graphics, financial mathematics, and many more.

The program trains students and scholars from across Stanford in mathematical modeling, scientific computing, and advanced computational algorithms at the undergraduate and graduate levels. Courses typically provide strong theoretical foundations for solving real-world problems and numerical computations to facilitate the application of mathematical techniques and theories. Training offered includes matrix computations, computational probability and combinatorial optimization, optimization, stochastics, numerical solution of partial differential equations, parallel computer algorithms, and new computing paradigms.

See Graduate Degrees for more information on the university’s basic requirements for the PhD degree.

Requirements

The Bulletin is the official record of Stanford program requirements. Following the degree requirements and deadlines listed below is required to maintain good academic standing in the CME-PhD program.

The coursework requirements for the ICME PhD total 57 units. Students are additionally required to take a minimum of 60 units of CME 400 (PhD Dissertation Research). The remaining 18 units required for the completion of the PhD can be a combination of CME 400 and additional elective courses.

Coursework completion deadlines: The deadlines for coursework requirements are listed below. Please note that year 2 requirements must be completed by Spring quarter ahead of advancing to candidacy:

Complete by end of Year 1 (Summer): First-Year Core, Research Rotations

Complete by end of Year 2 (Spring): Programming Core, Seminars

Complete by end of Year 3 (Summer): General Core, Programming Electives

Complete by end of program: Computational and Mathematical Electives, Specialized Depth Electives, Dissertation Research Units, Additional Units (Depth or Dissertation)

Academic year enrollment: ICME PhD students are required to take 8-10 units per quarter during the academic year.

Summer enrollment: First-year PhD students are required to enroll full-time during their first summer quarter. In years 2 and beyond, students work with their advisor and ICME student services to determine appropriate enrollment levels during summer quarter, and students may opt to pursue internship opportunities rather than enrolling.

Program Requirements

First-Year Core

CME 302, CME 306, and CME 307 must be completed during the first year of the CME-PhD program.

Complete all of the following course(s):
CME 302

Numerical Linear Algebra

3 Units
CME 306

Computational Methods of Applied Mathematics

3 Units
CME 307

Optimization

3 Units

First-year core classes must be taken for a letter grade.

Research Rotations

Complete nine units of research rotations in the first year.

This requirement is satisfied by securing a research rotation agreement with a faculty member and enrolling in CME391. Students may rotate with one or multiple faculty advisors. Research rotations aim to identify a suitable topic for PhD work and align with a faculty advisor by the summer of year one.

Earn at least 9 units in the following course(s):
CME 391

Ph.D. Research Rotation

1-10 Units

Programming Core

Students must complete CME 212 before advancing to candidacy at the end of year 2.

Complete all of the following course(s):
CME 212

Programming for performance

3 Units

Course must be taken for a letter grade.

Seminars

Complete three units of seminars before advancing to candidacy at the end of year 2. One unit must come from CME 300, one unit from either CME 500 or CME 510, and the remaining unit can come from any other listed seminar.

Complete 3 - 3 courses and earn 3 - 3 units in the following course(s):
CME 300

First Year Seminar Series

1 Units
CME 500

Departmental Seminar

1 Units
or
CME 510

Linear Algebra and Optimization Seminar

1 Units
APPPHYS 300

Department Colloquium

1 Units
or
BIOE 293

Bioengineering Department Colloquium

1 Units
or
CS 547

Human-Computer Interaction Seminar

1 Units
or
ENGR 298

Seminar in Fluid Mechanics

1 Units
or
GEOPHYS 385H

Subsurface Modeling and Characterization

1 Units
or
MED 216

Generative AI and Medicine

1 Units
or
MS&E 494

The Stanford Energy Seminar

1 Units
or
STATS 260A

Workshop in Biomedical Data Science

1-2 Units
or
STATS 260B

Workshop in Biomedical Data Science

1-2 Units
or
STATS 260C

Workshop in Biomedical Data Science

1-2 Units

Seminars may be taken for S/NC or letter grades (if offered). An NC grade will not satisfy the program requirements.

General Core

General Core classes must be completed by the end of year 3.

Complete all of the following course(s):
CME 303

Partial Differential Equations of Applied Mathematics

3 Units
CME 308

Stochastic Methods in Engineering

3 Units
or
CME 298

Probability and Stochastic Differential Equations for Applications

4 Units
or
MATH 236

Introduction to Stochastic Differential Equations

3 Units
CME 310

Combinatorial Optimization

3 Units

Courses must be taken for a letter grade.

Programming Electives

Programming Electives must be completed by the end of year 3. CS 149 is recommended.

Complete at least 1 of the following courses:
CS 149

Parallel Computing

3-4 Units
CS 217

Hardware Accelerators for Machine Learning

3-4 Units
CS 246

Mining Massive Data Sets

3-4 Units
CS 295

Software Engineering

3 Units
CS 315B

Parallel Computing Research Project

3 Units
CS 336

Language Modeling from Scratch

3-5 Units

Course must be taken for a letter grade. ICME PhD students entering our program with a CS undergraduate or MS degree may petition to replace one programming elective with another course. Approval is granted at the discretion of the faculty director and/or DGS. Petitions are not guaranteed approval.

Computational and Mathematical Electives

Computational and Mathematical Electives are courses with a strong computing or mathematical component. Students may select from the listed electives or petition for additional courses not on this list. These classes may be taken at any point in the program prior to advancing to TGR.

Complete at least 3 of the following courses:
CME 309

Randomized Algorithms and Probabilistic Analysis

3 Units
CME 334

Optimization Algorithms

3 Units
CME 364A

Convex Optimization I

3 Units
CME 364B

Convex Optimization II

3 Units
CS 224W

Machine Learning with Graphs

3-4 Units
CS 229M

Machine Learning Theory

3 Units
CS 238

Decision Making under Uncertainty

3-4 Units
MS&E 228

Applied Causal Inference with Machine Learning and AI

3 Units
MS&E 233

Game Theory, Data Science and AI

3 Units
STATS 300A

Theory of Statistics I

3 Units
STATS 300B

Theory of Statistics II

3 Units
STATS 300C

Theory of Statistics III

3 Units
STATS 315A

Modern Applied Statistics: Learning

3 Units
STATS 315B

Modern Applied Statistics: Learning II

3 Units

Courses must be taken for a letter grade.

Specialized Depth Electives

Specialized depth electives are courses determined in concert with the PhD advisor in order to develop in areas relevant to the student's research. The courses below are suggested options for specialized depth electives. Students may select from the list below or petition for additional courses not on this list. These classes may be taken at any point in the program prior to advancing to TGR.

Complete at least 4 of the following courses:
APPPHYS 229

Statistical Mechanics of Learning and Computation

3 Units
APPPHYS 237

Quantitative Evolutionary Dynamics and Genomics

3 Units
CME 262

Imaging with Incomplete Information

3-4 Units
CME 279

Computational Biology: Structure and Organization of Biomolecules and Cells

3 Units
CME 285

Computational Modeling in the Cardiovascular System

3 Units
CME 309

Randomized Algorithms and Probabilistic Analysis

3 Units
CME 330

Applied Mathematics in the Chemical and Biological Sciences

3 Units
CME 364A

Convex Optimization I

3 Units
CME 364B

Convex Optimization II

3 Units
CME 369

Computational Methods in Fluid Mechanics

3 Units
CS 205L

Continuous Mathematical Methods with an Emphasis on Machine Learning

3 Units
CS 221

Artificial Intelligence: Principles and Techniques

3-4 Units
CS 224N

Natural Language Processing with Deep Learning

3-4 Units
CS 224W

Machine Learning with Graphs

3-4 Units
CS 228

Probabilistic Graphical Models: Principles and Techniques

3-4 Units
CS 229

Machine Learning

3-4 Units
CS 229M

Machine Learning Theory

3 Units
CS 231N

Deep Learning for Computer Vision

3-4 Units
CS 234

Reinforcement Learning

3 Units
CS 236

Deep Generative Models

3 Units
CS 238

Decision Making under Uncertainty

3-4 Units
CS 246

Mining Massive Data Sets

3-4 Units
CS 251

Cryptocurrencies and blockchain technologies

3 Units
CS 264

Beyond Worst-Case Analysis

3 Units
CS 271

Algorithmic Foundations for Artificial Intelligence in Healthcare

3-4 Units
CS 328

Foundations of Causal Machine Learning

3 Units
CS 329S

Machine Learning Systems Design

3-4 Units
CS 331X

AI for Algorithmic Reasoning and Optimization

3 Units
CS 336

Language Modeling from Scratch

3-5 Units
CS 361

Engineering Design Optimization

3-4 Units
CS 448I

Computational Imaging

3 Units
ECON 273

Advanced Econometrics I

3-5 Units
ECON 274

Advanced Econometrics II

3-5 Units
EE 368

Digital Image Processing

3 Units
ENERGY 223

Subsurface Flow Simulation

3 Units
ENERGY 224

Advanced Subsurface Flow Simulation

3 Units
ESS 239

Data Science for Geoscience

3 Units
ESS 271

Climate Models and Data

3 Units
MATH 205A

Real Analysis

3 Units
MATH 205B

Real Analysis

3 Units
MATH 205C

Topics in Harmonic Analysis

3 Units
MATH 210A

Modern Algebra I

3 Units
MATH 210B

Modern Algebra II

3 Units
MATH 215A

Algebraic Topology

3 Units
MATH 215B

Differential Topology

3 Units
MATH 236

Introduction to Stochastic Differential Equations

3 Units
MATH 238

Mathematical Finance

3 Units
MATH 273A

Topics in Mathematical Physics

3 Units
MATH 273B

Topics in Mathematical Physics

3 Units
MATH 275A

Topics in Applied Math: Quantum Algorithms for Scientific Computation

3 Units
MATH 275B

Topics in Applied Math: A world of flows I

3 Units
MATH 275C

Topics in Applied Mathematics

3 Units
MATSCI 331

Computational Materials Science at the Atomic Scale

3 Units
ME 334

Advanced Dynamics, Modeling and Analysis

3 Units
ME 335A

Finite Element Analysis

3 Units
ME 336

Discontinuous Galerkin Methods for Fluid-Flow Simulations

3 Units
ME 344

Introduction to High Performance Computing

1 Units
ME 346A

Introduction to Statistical Mechanics

3 Units
ME 351A

Fluid Mechanics

3 Units
ME 351B

Fluid Mechanics

3 Units
ME 361

Turbulence

3 Units
ME 408

Spectral Methods in Computational Physics

3 Units
ME 470

Uncertainty Quantification

3 Units
MS&E 228

Applied Causal Inference with Machine Learning and AI

3 Units
MS&E 232H

Introduction to Game Theory (Accelerated)

3 Units
MS&E 235B

Reinforcement Learning: Behaviors and Applications

3 Units
MS&E 321

Stochastic Systems

3 Units
MS&E 323

Stochastic Simulation

3 Units
MS&E 325

Diffusion, Control and Optimal Transport

3 Units
MS&E 328

Foundations of Causal Machine Learning

3 Units
MS&E 334

Topics in Social Data

3 Units
MS&E 338

Aligning Superintelligence

3 Units
MS&E 349

Financial Statistics

3 Units
PHYSICS 234

Advanced Topics in Quantum Mechanics

3-4 Units
STATS 203

Regression Models and Analysis of Variance

3 Units
STATS 217

Introduction to Stochastic Processes I

3 Units
STATS 218

Introduction to Stochastic Processes II

3 Units
STATS 219

Stochastic Processes

4 Units
STATS 300A

Theory of Statistics I

3 Units
STATS 300B

Theory of Statistics II

3 Units
STATS 300C

Theory of Statistics III

3 Units
STATS 305A

Applied Statistics I

3 Units
STATS 305B

Applied Statistics II

3 Units
STATS 305C

Applied Statistics III

3 Units
STATS 318

Modern Markov Chains

3 Units
STATS 354

Generalization and Causality in Biohealth

1-3 Units
STATS 370

Bayesian Statistics

3 Units

Courses must be taken for a letter grade.

Dissertation Research

Enroll in dissertation research under your PhD advisor. Begin enrolling in CME 400 the first quarter after you have aligned with your advisor. This requirement is satisfied by enrolling in a minimum of 60 units of CME 400 across the duration of your PhD program.

Earn at least 60 units in the following course(s):
CME 400

Ph.D. Research

1-15 Units

Additional Units: Depth and/or Dissertation

To fulfill the remaining 18 units of your PhD program, please complete any combination of CME 400 and/or any classes listed in the Specialized Depth Electives section.

Advancement to Candidacy

Qualifying Exams

Pass the ICME qualifying exams at the end of the first year. These written exams are held in June and September. Students can attempt each exam twice. The qualifying exams cover material from CME 302, CME 306, and CME 307. 

A student who fails two or more qualifying exams after two attempts would not be eligible to advance to candidacy. Such students may be allowed to complete a CME MS degree. A student who fails one qualifying exam after two attempts will be required to undertake remedial action under the supervision of the failed exam’s faculty member. The typical corrective action is for the student to complete a literature survey on the exam topic. The survey scope should be agreed upon with the faculty member responsible for the exam. The faculty member is responsible for certifying the completion of this requirement, which must be concluded before the declaration of candidacy. 

Presentation of Research

Present at a research forum organized by the institute during the second year of the PhD. Students will showcase highlights from research rotations undertaken during the first year or present their PhD work’s initial steps. Students will prepare in consultation with their PhD advisor and receive faculty feedback following the research forum.

Advisor Alignment

Align with a permanent research advisor no later than the summer quarter of year one. Students must submit the signed advisor agreement opens in a new tab to ICME student services before the start of their second year.

Aligning with an advisor is an essential milestone in the PhD program and is required for making satisfactory degree progress and maintaining good standing.

Upon aligning with an advisor, students on the ICME first-year fellowship transition from department fellowship to advisor-provided funding. Therefore, students must select an advisor who can financially support them. 

PhD students who do not align with an advisor by summer quarter of year one are subject to dismissal from the program.

PhD students may need to change advisors after initial alignment due to unexpected circumstances. Since an advisor is necessary for continued funding and good standing, please consult the Director of Student and Academic Services or Director of Graduate Studies as early as possible to ensure proper coordination and uninterrupted support. Additional guidance is available in this resource opens in a new tab.

Dissertation proposal

It is strongly recommended that students submit a dissertation proposal to their advisor in conjunction with advancing to candidacy. The student and advisor will determine the process and format of this proposal. Students should indicate the dissertation proposal submission in section D of the university candidacy form.

Candidacy forms

Submit the completed candidacy opens in a new tab form to the ICME student services team by the end of spring quarter in year two. The student’s dissertation advisor must review and sign the form.

University policy requires all doctoral students to declare candidacy by the end of the sixth quarter in residence, excluding summers. Candidacy expires five years from the date of submission of the candidacy form, rounded to the end of the quarter. In exceptional cases, the department may extend a student’s candidacy but is under no obligation to do so.

PhD candidacy forms are approved automatically if all the pre-candidacy requirements are satisfied. Exceptional circumstances (such as, but not limited to, extended sickness, visa problems, or leave of absence) will be considered by the ICME director, who will be responsible for making necessary adjustments to the process, if applicable and within the bounds of university policies. Students should petition the ICME director and student services in writing to initiate the process and detail their circumstances.

In all other cases where a student has not met the candidacy requirements by the end of year two, the ICME faculty steering committee will review the student’s progress holistically and provide a recommendation. Students are encouraged to submit a short assessment of their progress and provide further evidence that might be useful to the committee.

Reading Committee

In consultation with the advisor, the student must form a Reading Committee. A Reading Committee Form opens in a new tab must be completed and filed with the ICME student services office before the approval of TGR status. University policy requires the appointment of a principal dissertation advisor currently on the Academic Council. Former Academic Council members, emeritus Academic Council members, or non-Academic Council members can serve as co-advisor. Please consult the Graduate Academic Policy opens in a new tab for guidelines on how to form your Reading Committee.

Green Light Meeting

At least six months before the planned PhD Defense, the student and the Reading Committee must set a Green Light Meeting. In this meeting, a final dissertation proposal will be reviewed and discussed. The student will present a draft PhD thesis outline indicating prior work status and plans for additional work.

The presentation should not be a practice thesis defense but a chapter-by-chapter review of the completion status for each part of the expected PhD thesis. A month-by-month schedule for the time remaining until the PhD defense should be presented at this meeting. The purpose of the Green Light Meeting is to ensure all members of the Reading Committee are familiar with the plans for the completed PhD thesis and agree with the proposed content and schedule. Detailed instructions are provided on the ICME Intranet.

Oral Exam

Oral Exam

Any time after completing an acceptable draft of the dissertation, with the approval of the advisor and Reading Committee, the student may schedule the University Oral Examination (Dissertation Defense). To do so requires the completion of the University Oral Examination form opens in a new tab. Once the examiners, the date, and the time for the oral examination have been established, this form must be submitted to the ICME student services office for approval at least two weeks before the proposed examination date. It is the student’s responsibility to schedule the time and day of the exam and ensure that all examiners can attend. The student is also responsible for ensuring that the exam Chair is given the University Oral Examination Form before the exam starts. If the Chair of the exam does not have the form before the start of the exam, the exam is invalid.

The student must be registered during the term the University Oral Examination is taken. The student is encouraged to email an announcement for the public portion of their Dissertation Defense to the ICME student services office two weeks before the defense to advertise it. For information on forming your Oral Exam Committee, see Stanford’s Graduate Academic Policy opens in a new tab.

 The Oral Exam can be up to three hours. The student should consider this when scheduling the time with the Oral Exam Committee. The Oral Exam Chair should submit the examination results to the ICME student services office immediately following the exam. See further information on the ICME Intranet and the Graduate Academic Policy page.

Dissertation

Dissertation

ICME PhD students are required to complete a dissertation. Consult the university's page on dissertation information opens in a new tab for rules governing the dissertation format, fees, forms, and deadlines.