SPDE-Visuals
Image collection
This folder contains a variety of images for talks, lectures, and presentations on SPDEs and related topics. Most were generated using MidJourney (denoted by the prefix MJ or my username on Discord chenle02). One wide illustration, Nano_Banana_Pro_Classical_to_Singular_SPDEs_regularity_barrier.png, was generated by Gemini 3 using Nano Banana Pro, and two related images, Sora_Classical_to_Singular_SPDEs_landscape_v1.png and Sora_Classical_to_Singular_SPDEs_landscape_v2.png, were generated by Sora@OpenAI. Filenames are based on the prompts used to create the images.
Table of contents
- Stochastic heat equation (SHE)
- Cosmic and universe themes
- Natural and environmental themes
- Mathematical and theoretical concepts
- Examples of stochastic wave equation
- Usage
Stochastic heat equation (SHE)
These images are visual representations or interpretations of the stochastic heat equation (SHE), parabolic Anderson model, etc. :
- chenle02_Stochastic_heat_equation.png
- chenle02_Chaotic_stochastic_heat_flow_on_a_torus.png
- chenle02_Intermittency_parabolic_Anderson_model_Brownian_motion.png
- MJ_SHE.png
- MJ_SHE_Sphere.png
- MJ_SHE_Torus.png
The following images are used to show my favorite initial condition: the Dirac delta initial data.
The following examples illustrate intermittency and are taken from Zeldovich’s book Almighty Chances:
- chenle02_Mushrooms_in_clusters_in_a_forest_with_sun_shades_ultr.png
- chenle02_In_a_dense_forest_there_is_a_folk_where_two_roads_dive_fbb81936-8ba4-41b4-af9a-5872d385313d.png
- MJ_Mushroooms_Forest.png
- MJ_Mushroooms_Forest_4pics.png
Cosmic and universe themes
Exploring cosmic forces, the universe, and other celestial concepts:
- chenle02_A_person_in_front_of_the_whole_universe_gallaxies.png
- chenle02_Cosmic_forces_beyond_comprehension.png
- chenle02_Was_it_nuclear_explosionor_cosmic_forces_beyond_comprehension.png
- chenle02_A_homesick_guy_looking_back_to_Earch_from_Mars_cinemar.png
- chenle02_Standing_on_the_planet_Mars_looking_at_home_the_earth.png
- chenle02_Imagine_a_massive_object_such_as_a_black_hole_or_neutr_335e7ef3-5235-4cda-8233-f3feb9eac5c5.png
- MJ_Curious_person_Facing_Universe1.png
- MJ_Curious_person_Facing_Universe2.png
- MJ_Evolution_Universe_4pics.png
Natural and environmental themes
Nature-inspired images, including forests, mushrooms, and lightning:
- chenle02_Show_rain_drops_on_a_small_pond_in_a_foggy_raining_day.png
- chenle02_Lightningclouds_dramatic_light_zigzag_highly_detailed.png
- MJ_Folk_in_Forest.png
- MJ_Folk_in_Forest_more_realistic.png
- MJ_Growing_Snow_Sufface.png
Mathematical and theoretical concepts
Visualizations inspired by mathematical ideas and models:
- chenle02_Law_of_large_numbers_and_central_limit_theorem.png
- chenle02_Demonstrate_the_turbulent_chaotic_stochastic_heat_flow_1a0001f2-80fa-4f5f-bdec-4127c89b326d.png
- MJ_Chaos_Intermittency_SPDEs1.png
- MJ_Chaos_Intermittency_SPDEs2.png
- MJ_SWE_Chaos_Propogation.png
Polymer and energy-flow inspired landscapes (generated by Sora@OpenAI, 2025‑11‑13):
The following three images contrast classical SPDEs with singular SPDEs across the regularity barrier (one wide illustration by Nano Banana Pro / Gemini 3 and two cinematic variants by Sora@OpenAI):
- Nano_Banana_Pro_Classical_to_Singular_SPDEs_regularity_barrier.png
- Sora_Classical_to_Singular_SPDEs_landscape_v1.png
- Sora_Classical_to_Singular_SPDEs_landscape_v2.png
Examples of stochastic wave equation
The following is an example from Walsh’s 1986 notes, which describes a one-dimensional stochastic wave equation—the vibration of a guitar string in a sandstorm:
Short companion videos illustrating the same scene (generated by Sora@OpenAI):
Inline playback on GitHub (may depend on browser support):
Usage
Feel free to use any images and videos in this folder for talks, lectures, papers, and creative projects—no formal acknowledgment is required.
If you would like to express gratitude, a brief note such as
Figures inspired by the SPDE-Visuals repository (Le Chen, Auburn University, https://spdengine.github.io/SPDE-Visuals/).
or a link to the GitHub/GitHub Pages project is very much appreciated. When appropriate, you may also credit the tools used (e.g., MidJourney, Gemini 3 / Nano Banana Pro, Sora@OpenAI). For questions or comments, you can reach me at lzc0090@auburn.edu or visit https://webhome.auburn.edu/~lzc0090/index.html.
For further exploration and downloads, visit the main repository: SPDE-Visuals.