Data-Driven Modeling for Scientists & Engineers (3b/6): Nonlinear Heat Equation & Machine Learning

In this lecture, we explore the second-order logistic rate law—a model that captures both production and consumption dynamics in chemistry, biology, and ecology. We generate a large-scale dataset and feed it into a gray box, physics-informed neural network to recover the underlying reaction kinetics from data alone. We connect this model to autocatalysis (including the Belousov–Zhabotinsky reaction), heterogeneous catalysis, epidemic spread, population dynamics, and bacterial biofilms, showing how the same mathematical form arises across disciplines. You’ll also see practical training challenges—balancing PDE residual loss with data loss, weighting in time to control error distribution, and interpreting learned rate laws. This video is ideal for chemical engineers, applied mathematicians, computational scientists, and biophysicists who want to combine experimental data, PDEs, and ML for interpretable, transferable modeling. Data-driven modeling and discovery These videos were inspired in part during my initial brainstorming of the materials to teach in my graduate course at Stanford University on the subject of Applied Mathematics for engineers.  I've felt that many scientists and engineers are taught abstract, analytical concepts in math, but have very little experience dealing with actual numbers, large datasets, matrices, noisy data.  But in reality, researchers deal with messy & large datasets ALL OF THE TIME.  My goal was to get students to actually deal with messy & large data to help them in their PhD theses.  I’m sharing these here in case they’re helpful to students, educators, or just anyone who is curious about the (over-promised) hype surrounding ML, AI, etc.  What You'll Learn (6-ish lectures):  • Different types of modeling approaches (L1) • Using machine learning and neural nets to solve PDEs (L2a, L2b) • Using neural nets to solve nonlinear PDEs with unknown sources (L3a, L3b) • Sparse regression (L4) • Concrete example problems using data-driven methods (L5) • Hidden variables, embeddings, and observability (L6) Why This Matters:  Modern science and engineering are overflowing with data—but also overflowing with tools. In these lectures, we explore different ways to use data to develop models and discover unmodeled dynamics. You’ll learn how to extract interpretable, physics-aware models from data, and how to bridge the gap between computational power and human insight. Whether you're working on fluid mechanics, reaction kinetics, or biological networks, these lectures are intended to get you thinking about how data-driven modeling could turn messy observations into meaningful science. About Me:  I’m a professor of Chemical Engineering at Stanford running a research lab on soft condensed matter and fluid mechanics. I'm broadly interested in nonlinear dynamics and many-body collective phenomena ... which includes just about everything in nature. I post videos on engineering, soft matter physics, data-driven models, and the art of science communication. Think of this as my way of opening the classroom to a wider audience. Outside of research, I have had a strong passion for public speaking since high school, taking speech courses in college and competing in speech contests in Toastmasters International (a professional organization to improve public speaking and leadership skills) for several years as a PhD student. More recently, as a professor and educator, I have channeled my passion for speaking towards science education and technical communication. I have always believed that effective science communication can make broad impacts to society by building public trust in science, promoting data-driven decisions in government and industry, and improving the accessibility of science to all communities. I look forward to continue working on effective science communication skills and storytelling techniques with students and researchers in my career.  My contents are personal and do not represent Stanford University.