PDE Solver Guide

Overview

This chapter covers the full workflow on the PDE Solver page. It fits these scenarios:

You are on the PDE Solver page but unsure about the order of the different regions.
You want to complete a full loop from “select an equation” to “submit a task” to “view the result”.
You plan to edit a custom equation or use AI chat to understand parameters and task states.

Topics: Select Equation, Create Custom Equation, Edit Equation, Parameter Configuration, Submit Task, Monitor Task, View Task Details, AI Chat, Supported PDEs.

Page Overview

The PDE Solver page is a multi-region workbench with relatively high information density.

Annotated PDE solver workbench

Numbers in the figure correspond to:

Equation tree: browse equations by group and select the target one.
Main content area (left): shows formulas, README, and equation overview.
Parameter panel: located on the right side of the main content area; configure solver parameters by group.
Bottom task monitor: view task information, the quantum circuit, and results.
AI chat panel: a dedicated right-side panel for asking questions about equations, parameters, or tasks.

The Task Manager entry is at the bottom of the left navigation bar. Click it to expand and review all historical submitted tasks.

When equation data is loading, the page shows a loading indicator; if loading fails, it shows an error with a retry button.

A simple mental model for the page — four blocks:

Left: decide “which equation to run”.
Center: view equation content (left half) and configure solver parameters (right half).
Right: use AI chat for understanding and assistance.
Bottom: check “how the run is going and what the result is”.

Select Equation

The left region is used to select an equation:

Equations are shown as groups. On entering the page, the first equation of the first group is selected by default, and its details and parameters are loaded automatically.
Click a group to expand or collapse its equation list.
Clicking a specific equation switches the page to that equation, loading the matching details and parameters.

The top of the equation list provides a search box — typing a keyword quickly filters equations, which helps when the list is long.

If equation data failed to load, use the in-page retry button instead of refreshing the whole browser.

Create Custom Equation

If the built-in equations do not meet your needs, click the + button on the right of the equation list header to open the new-equation dialog and create a custom equation.

New equation dialog

The dialog contains three fields:

Equation name: the human-readable name shown in the UI; used to identify the equation in the list.
Equation key: the unique internal key used by the program; only letters, digits, and underscores are allowed.
Group: assigns the equation to an existing group; click the dropdown to pick an existing group, or click New Group at the bottom to create a new category.

Group dropdown

After filling in the fields, click Create to finish; the new equation appears under the matching group in the list.

Note: Neither the equation key nor the group name can duplicate an existing one, or creation will fail.

Delete a custom equation / group: When hovering on a custom equation or custom group, a delete button appears on the right; click it and confirm in the dialog to remove it. Built-in equations and built-in groups cannot be deleted.

When creating a custom equation makes sense:

None of the built-in equations matches your needs.
You need a long-term set of custom equation configurations instead of a one-off parameter tweak.

Edit Equation

Two ways to enter the equation editing state:

Automatically: after creating a custom equation, the system enters edit mode automatically.
Manually: click an equation in the left tree, then click the Settings button at the top-right of the parameter panel.

After entering edit mode, the right-side parameter panel collapses automatically to give the editing area more room.

Note: Built-in equations only allow editing the algorithm file; custom equations support editing the JSON equation config (equation definition, boundary conditions, initial conditions, and so on).

Annotated custom equation editor

Numbers in the figure correspond to:

Content tabs: switch among formula, README, algorithm file, or JSON config.
Action buttons: code/preview toggle, exit edit, and save.
Editing area: shows the content for the active tab; when editing JSON, the left side is typically the group navigation and the right side is the form for the current group.

After editing, click Save in the top-right to apply the changes; the updates sync to the equation parameters and formula display. Clicking the X button closes the editor without saving, and the next time you open it, the last saved content is restored.

When the active tab is a .json config file of a custom equation, a Code / Preview toggle also appears in the top-right. Code mode edits the JSON directly; Preview mode views and edits the config via a structured form.

Annotated JSON code editor and preview toggle

Numbers in the figure correspond to:

Code / Preview toggle area: switches the JSON between code mode and preview mode; Save and Close buttons also live here.
JSON code editor: edit the JSON config directly.

In Code mode, you can copy the JSON sample from the README into the editor and adjust fields as needed. Notes:

Check name, the equation type, and each group field first. If you save the sample name as-is, the current custom equation’s name is overwritten along with it.
When the JSON is invalid, the Save button is disabled, and clicking Preview reports an error and stays in code mode. Fix the JSON before continuing.

When using the editor, keep “editing the equation” and “tuning parameters” separate:

Editing the equation: changes the equation definition itself, boundary conditions, initial conditions, or the algorithm file.
Tuning parameters: keeps the equation definition intact and only adjusts the input values for this run.

For a quick trial run, tuning parameters is usually enough; only enter the editor when the equation definition itself needs to change.

Parameter Configuration

After selecting an equation, the parameter panel appears on the right.

Annotated parameter panel

Numbers in the figure correspond to:

Top info bar: shows the current equation name; the top-right offers a settings toggle and a save action.
Parameter area: expand each group to fill in the solver parameters for the current equation.

Two common buttons sit at the top-right of the parameter area:

Settings button: switches to the editing view for the current equation. Built-in equations enter the algorithm-file editor; custom equations enter the JSON equation-config editor.
Save button: saves the current in-panel changes for the selected equation. If you switch to another equation before saving, the current changes are lost; after a successful save, the next time you open this equation, the saved parameters are used.

Recommended order:

Pick the target equation first.
Review each default value in the parameter area on the right.
Modify parameters based on the task’s needs.
If the page shows a validation hint, fix it before continuing; while unfixed, submission is blocked and the submit button is disabled.

If you want to keep the current configuration next time you open the same equation, save first, then run.

Submit Task

The top-right corner of the page provides the run entry. Flow:

Submit task dialog

Complete equation selection and parameter configuration first.
Click the run entry.
Fill in the task name in the dialog.
Submit the task.

Give the task a name that distinguishes its purpose, such as one including the equation name, experiment goal, or date; this makes it easier to find later in task details and the result list.

Monitor Task

The task monitor area at the bottom is linked to the current task or the current equation’s tasks, showing execution state and output.

Task monitor panel

Task info: basic information and run log of the current task, useful for checking status, debugging errors, or confirming progress.
Quantum circuit: the quantum circuit generated for the current task.
Result: the task’s run result; result images and metrics are shown here.

When the result tab shows a result image, hovering on the result area reveals floating action buttons on the right:

Download button: download the current result image or file.
Fullscreen button: enter fullscreen mode for a larger detail view.

Annotated result actions

Numbers in the figure correspond to:

Download button: download the current result image or file.
Fullscreen button: enter fullscreen mode.

In fullscreen mode, a close button is provided at the top-right; the bottom toolbar supports zoom in, zoom out, checking the current zoom ratio, and downloading the current result content.

Annotated result fullscreen

Numbers in the figure correspond to:

Close button: exits fullscreen and returns to the result area.
Bottom toolbar: supports zoom in, zoom out, checking the current zoom ratio, and downloading the current result content.

The monitor area defaults to the most recently submitted task and follows the currently active task; switching equations or submitting again updates the bottom content in sync.

For first-time use of the PDE Solver, complete the simplest equation selection and task submission first, then observe how the bottom monitor area changes.

View Task Details

Click the Task Manager entry at the bottom-left to open the task detail dialog, where you can review all historical tasks and their run results.

Annotated task detail dialog

Numbers in the figure correspond to:

Search & filter bar: search by task name or filter by status.
Task list: every historical task; click an item to switch the detail view.
Detail area on the right: shows the selected task’s status, parameter configuration, and run result.

When the detail area shows a result image, hovering on the result area reveals floating action buttons on the right.

Annotated task detail result actions

Numbers in the figure correspond to:

Download button: download the current result image or file.
Fullscreen button: enter fullscreen mode for a larger detail view.

The Edit button at the top-right of the detail area lets you backfill the task’s parameters into the right-side parameter panel, which helps when you want to tweak and resubmit based on existing parameters.

When task details fit best:

You need to review historical tasks, not just the latest run.
You want to compare parameter differences and outputs across tasks.
You want to resubmit from a modified copy of a past task.

AI Chat

The right side of the PDE Solver page is a dedicated AI chat panel. You can ask about the current equation’s physical meaning, parameter configuration, or task results at any time without leaving the page.

Common PDE-specific questions: the equation’s applicable scenarios and parameter meanings, why a task failed and what to investigate, how to interpret results and how to tune the next run.

For full usage details — entry points, chat modes, image attachments, and model settings — see the AI Chat Guide.

Suggestions

The PDE Solver page has many regions — on your first visit, scan in the order “select on the left, configure on the right, check tasks at the bottom”.
On the first run, complete the most basic equation selection and task submission, then get familiar with the different parameter regions step by step.
If you only want to confirm that the page works, skip editing a custom equation at the start; completing a minimal task loop is more important.

Supported PDEs

Heat Equation

1D Forward Heat Equation

\frac{\partial u}{\partial t} = a \frac{\partial ^{2} u}{\partial x ^{2}} + f (x)

It describes the process of heat conduction from high to low temperature in a medium, reflecting the diffusion law of energy through molecular thermal motion. A larger diffusion coefficient $a$ leads to faster heat conduction. It is applicable to scenarios such as temperature field calculation in metal heat treatment, thermal insulation design of building walls, and thermal loss analysis of semiconductor chips, and can predict the temperature distribution change of an object during heating or cooling.

1D Backward Heat Equation

\frac{\partial u}{\partial t} = - a \frac{\partial ^{2} u}{\partial x ^{2}} + f (x)

It describes a mathematically time-reversed heat conduction process, representing the inverse diffusion of energy. A larger diffusion coefficient $a$ leads to faster heat conduction. It is applicable to scenarios in theoretical and inverse problems such as reconstructing initial temperature fields in heat treatment processes, back-calculating thermal histories in building insulation analysis, and solving certain inverse problems in semiconductor thermal management.

1D Variable-Coefficient Forward Heat Equation

\frac{\partial u}{\partial t} = a (x) \frac{\partial ^{2} u}{\partial x ^{2}} + f (x), a (x) = 1 + cos (2 π x / L)

It describes the process of heat conduction from high to low temperature in a medium, reflecting the diffusion law of energy through molecular thermal motion, where a larger diffusion coefficient $a (x)$ leads to faster heat conduction locally. It is applicable to scenarios such as temperature field calculation in metal heat treatment, thermal insulation design of building walls, and thermal loss analysis of semiconductor chips, and can predict the temperature distribution change of an object during heating or cooling, accounting for spatial or temporal variations in the material’s thermal conductivity.

2D Forward Heat Equation

\frac{\partial u}{\partial t} = a_{1} \frac{\partial ^{2} u}{\partial x ^{2}} + a_{2} \frac{\partial ^{2} u}{\partial y ^{2}} + f (x, y)

It extends heat conduction to a two-dimensional plane, accounting for temperature gradients along both $x$ and $y$ . Typical uses include thin plates, chip/PCB thermal analysis, and crustal heat-flow modeling.

2D Backward Heat Equation

\frac{\partial u}{\partial t} = - a_{1} \frac{\partial ^{2} u}{\partial x ^{2}} + - a_{2} \frac{\partial ^{2} u}{\partial y ^{2}} + f (x, y)

It extends heat conduction to a two-dimensional plane, accounting for temperature gradients along both $x$ and $y$ . Typical uses include reconstructing initial temperature fields in thin plates, chip/PCB thermal analysis, and crustal heat-flow modeling.

Advection Equation

1D Advection Equation

\frac{\partial u}{\partial t} = a \frac{\partial u}{\partial x}

It describes the transport process of substances (such as pollutants and water vapor) with fluid motion. The velocity $a$ determines the migration direction and rate of the physical quantity. In environmental science, it can simulate the diffusion path of river pollutants with water flow. In meteorology, it is used to analyze the transport of water vapor in the atmospheric troposphere. In petroleum engineering, it describes the flow law of crude oil in the reservoir.

Black-Scholes Equation

1D Black-Scholes Equation

\frac{\partial u}{\partial t} + a (r - \frac{σ ^{2}}{2}) \frac{\partial u}{\partial x} + \frac{σ ^{2}}{2} \frac{\partial ^{2} u}{\partial x ^{2}} - r u = 0

Core model for European option pricing. Using the risk-free rate $r$ and volatility $σ$ , the PDE governs the time evolution of the option value $u (t, x)$ (with $x$ typically the log-price). The factor $a$ allows optional scaling in our framework.

The above log-price form is obtained from the classical price-space PDE via $S = e^{x}$ and $u (t, x) = V (t, e^{x})$ :

\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} + r S \frac{\partial V}{\partial S} - r V = 0

Elastic Wave Equation

1D Elastic Wave Equation

ρ \frac{\partial ^{2} u}{\partial t ^{2}} = (λ + 2 μ) \frac{\partial ^{2} u}{\partial x ^{2}} + f (x)

2D Elastic Wave Equation

ρ \frac{\partial ^{2} u ( x )}{\partial t ^{2}} = (λ + μ) \nabla (\nabla \cdot u (x)) + μ \nabla^{2} u (x) + f (x, y) v_{i} = \frac{\partial u _{i}}{\partial t}, σ_{ij} = λ δ_{ij} (\nabla \cdot u) + μ (\frac{\partial u _{i}}{\partial x _{j}} + \frac{\partial u _{j}}{\partial x _{i}})

They both describe the propagation of mechanical waves (such as seismic waves and ultrasonic waves) in an elastic medium. The Lamé coefficients $λ$ and $μ$ together with the medium density $ρ$ determine the wave speed. In seismology, it is used to invert the internal structure of the earth. In non-destructive testing, ultrasonic waves are used to detect cracks in metal components. In petroleum exploration, seismic wave data are used to locate underground oil and gas reservoirs.

Burgers Equation

1D Burgers Equation

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = ν \frac{\partial ^{2} u}{\partial x ^{2}}

It describes the wave behavior under the combined action of nonlinear convection and viscous diffusion. It is a basic model for simulating the formation of shock waves in fluid mechanics. The parameter $ν$ represents the kinematic viscosity coefficient, which controls the intensity of the diffusion effect. When $ν = 0$ , the equation degenerates into the inviscid Burgers’ equation, and discontinuous shock waves may appear in the solution. When $ν > 0$ , the viscous effect makes the shock wave transition to a smooth traveling-wave solution.

2D Burgers Equation

⎩ ⎨ ⎧ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = 0 \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = 0

The 2D inviscid Burgers’ equation is a fundamental model in fluid dynamics used to describe the convection of velocity fields and the formation of shock waves. Due to its nonlinear structure, even smooth initial conditions can develop discontinuities (shocks) in finite time. This algorithm treats the velocity fields $(u, v)$ as conserved quantities and applies the level-set method to linearize the dynamics in a higher-dimensional space.

Traffic Flow Equation

1D Traffic Flow Equation

\frac{\partial u}{\partial t} + f^{'} (u) \frac{\partial u}{\partial x} = 0

For the classical traffic flow model, the flux function is typically $f (u) = u (1 - u)$ .

This equation originates from the classical Lighthill-Whitham-Richards (LWR) model, which employs scalar conservation laws to describe the evolution of vehicle density and the emergence of traffic congestion. Due to the nonlinearity of $f (u)$ , characteristics can intersect, leading to shock formation in finite time, even for smooth initial data. By combining a level set preprocessing method with Schrödingerization, this complex nonlinear phenomenon can be transformed into a linear unitary evolution suitable for quantum simulation.

Schrodinger Equation

2D Time-Dependent Schrodinger Equation

i \frac{\partial}{\partial t} ψ (t, x, y) = (- \frac{\nabla ^{2}}{2} + V (x, y)) ψ (t, x, y)

It describes the quantum mechanical evolution of a wave function on a plane. It is a basic model for simulating electron dynamics in two-dimensional materials and quantum dots in quantum computing. The potential V(x,y) defines the external forces, such as barriers or wells. A vanishing potential (V=0) results in a freely spreading wave packet, while a strong potential leads to localized, standing-wave eigenstates. The relative strength of the kinetic energy term (with m) and the potential $V$ dictates whether the behavior is wave-like, particle-like, or exhibits quantum tunneling.

Helmholtz Equation

1D Helmholtz Equation

- Δ u - k^{2} u = f, x \in (0, L)

It describes the propagation of time-harmonic waves in one-dimensional systems. It is a fundamental model for analyzing steady-state wave phenomena, such as resonance in acoustic cavities, electromagnetic waveguides, and quantum mechanical bound states. The parameter $k$ represents the wave number, which is proportional to the frequency of the harmonic oscillation and determines the spatial scale of the wave. When $k = 0$ , the equation simplifies to a Poisson equation.

Multiscale Equation

1D Multiscale Transport Equation

ϵ \partial_{t} f (v) + v \cdot \nabla_{x} f (v) = \frac{1}{ϵ} (\frac{σ _{S}}{S} \int_{Ω} f (v^{'}) d v^{'} - σ f (v)) + ϵ Q

It describes the kinetic behavior of particles in a one-dimensional geometry under a multiscale regime. It is a fundamental model for simulating radiation transport in shielding materials, neutron diffusion in nuclear reactors, and heat transfer in optically thick media. The scaling parameter $ϵ$ represents the ratio of the mean free path to the characteristic system length, distinguishing between collision-dominated ( $ϵ ≪ 1$ ) and free-streaming ( $ϵ ≫ 1$ ) regimes. When $ϵ \to 0$ , the equation asymptotically reduces to a diffusion equation. The source term $Q$ governs whether the solution behaves diffusively, ballistically, or in a transition regime between the two.

1D Multiscale Elliptic Equation

- \frac{d}{d x} (A_{ϵ} (x) \frac{d}{d x} u_{ϵ} (x)) = 1, A_{ϵ} (x) = (2 + cos (2 π x / ϵ)^{- 1})

It describes the steady-state diffusion in a one-dimensional heterogeneous medium with a rapidly oscillating coefficient $A_{ϵ}$ . The equation describes phenomena such as heat conduction in composite materials, groundwater flow in porous media, and electrostatic potentials in periodic structures. The parameter $ϵ$ represents the ratio of the microscopic period to the macroscopic length scale. As $ϵ \to 0$ , the solution $u_{ϵ}$ converges to a homogenized solution satisfying an effective (averaged) diffusion equation.

Maxwell Equation

1D Simplified Maxwell Equation

\frac{\partial E _{y}}{\partial t} + v \frac{\partial B _{z}}{\partial x} = - J_{y}, \frac{\partial B _{z}}{\partial t} + v \frac{\partial E _{y}}{\partial x} = 0,

These equations describe the propagation of electromagnetic waves in one dimension, representing a transverse electric (TE) or transverse magnetic (TM) mode simplification. Here, $E_{y}$ is the electric field component, $B_{z}$ is the magnetic field component, $v$ is the wave speed (typically $v = 1/ μ ϵ$ in a vacuum), and $J_{y}$ is the current density source term. The equations model phenomena such as signal transmission in coaxial cables, waveguides, and atmospheric electromagnetic wave propagation.

Stochastic Differential Equation

1D Ornstein-Uhlenbeck Process

d X (t) = a X (t) d t + r d W_{t}

This equation describes the dynamics of a stochastic process $X (t)$ . Deterministic drift term $a X (t) d t$ represents a deterministic trend proportional to the current state $X (t)$ , and stochastic diffusion term $r d W (t)$ represents random fluctuations, where $r$ is the volatility coefficient scaling the intensity of the noise, and $d W (t)$ is the increment of a standard Wiener process (Brownian motion).