Complete curriculum · Foundations through AI-driven methods
Differential Equations
A guided mathematical text spanning analytic reasoning, computational experiment, and agent-based scientific computing — from first principles to neural solvers and reinforcement learning for dynamical systems.
Begin with Level 0: Foundations, progress through ODEs and PDEs, then explore AI-driven methods and capstone projects.
Course Levels
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Level 0 — Foundations
Review of calculus, linear algebra, and an introduction to differential equations. Builds the prerequisite toolkit for the entire curriculum.
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Level 1 — Ordinary Differential Equations
First-order and second-order ODEs, systems of linear ODEs, and numerical ODE solvers with modeling applications and agent-based perspectives.
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Level 2 — Partial Differential Equations
Scalar PDE classification, numerical PDE methods (finite differences, finite elements), and stochastic and complex models.
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Level 3 — Computational & AI-Driven Differential Equations
Neural ODEs, physics-informed neural networks, neural operators, agent-based RL control for dynamical systems, and data-driven equation discovery.
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Level 4 — Projects, Applications & Research Pathways
Capstone projects: agent-based simulation, neural vs. classical solver benchmarks, and scientific machine learning for real data.
Curriculum Learning Objectives
- Formulate differential equations from physical, biological, and engineered systems.
- Solve canonical ODE and PDE classes analytically and verify assumptions computationally.
- Interpret phase portraits, stability, and asymptotic behavior with mathematical precision.
- Implement numerical methods (Euler, RK4, finite differences, finite elements) and critique approximation error in context.
- Design and train neural differential equation solvers, physics-informed neural networks, and neural operators.
- Frame dynamical systems as agent-environment loops: define state, action, reward, and learning objectives.
- Apply reinforcement learning to control ODE/PDE systems with stability and efficiency guarantees.
- Discover governing equations from data using sparse regression and symbolic methods.
- Benchmark classical and neural solvers on accuracy, speed, and generalization.
- Read, write, and present solution arguments in clear mathematical and computational style.
Code Example
A simple exponential decay model solved numerically and compared with its exact expression.
import numpy as np
def f(x, y):
return -2 * y
x = np.linspace(0, 3, 7)
y_exact = np.exp(-2 * x)y(x) = e-2x decreases monotonically toward 0 as x increases.