TFRD benchmark

TFRD Dataset

TFRD dataset is generated using the data genereator based on FEniCS, and you can download at BaiduPan（Password: tfrd）。

TFRD dataset describes the heat source system with $0.1m \times 0.1m$ square board, where the heat source system is discreted as $200 \times 200$ grid. It consists of the HSink, the ADlet, and the DSine sub-data. Each sub-data includes 10000 training samples, 20000 testing samples （Test 0 represents the randomly generated samples, and Test 1, 2, 3, 4, 5 describes special samples）:

• Generality

  Three typical heat source systems（HSink，ADlet，DSine）

• Reasonability

  The composition of TFRD is reasonable（includes 10000 training samples, 20000 testing samples divided into six different special samples）

• Diversity

  Provide diversified boundary conditions, components as well as evaluation metrics

HSink Sub-data

Heat-source systems with heat sink for heat dissipation where the width is set to 0.01m with a constant temperature valued 298K （Dirichlet boundary）.

Training Sample	Training Sample

Test 0	Test 1	Test 2
Test 3	Test 4	Test 5

ADlet Sub-data

Heat-source systems with all different Dirichlet boundary conditions for heat dissipation where one boundary is set to sine-wave distribution and the others are set to constant temperature valued 298K.

Training Sample	Training Sample

Test 0	Test 1	Test 2
Test 3	Test 4	Test 5

DSine Sub-data

Heat source systems with one sine-wave distributed boundaries for dissipation. All the other three boundaries are adiabatic （Neumann BC）.

Training Sample	Training Sample

Test 0	Test 1	Test 2
Test 3	Test 4	Test 5

Data Description

The sample is saved as mat format.

Example*.mat

• 'u'：{ndarray:200,200}，temperature information，ranging from 298-
• 'F'：{ndarray:200,200}，component information of the system，ranging from 0 to 30000
• 'u_obs'：{ndarray: 200 , 200 }，temperature information of monitoring information，ranging from 298-
• 'u_pos'：{ndarray: 200 , 200 }，position information of monitoring inforation，selected from {0, 1}，1 describes the monitoring points and 0 represents other area without monitoring points
• 'xs'，'ys'，'zs'：{ndarray: 200 , 200 }，x, y, z coordinates information，ranging from 0 to 0.1

Evaluation Metrics

mean absolute error（MAE）

Mean absolute error（MAE）measures the mean value of absolute error of the predicted temperature field.

$${E_{MAE} } = \frac{1}{ {\left| \Omega \right|} }\sum\limits_{({x_i},{y_j}) \in \Omega } {\left| {T({x_i},{y_j}) - T'({x_i},{y_j})} \right|}$$

Maximum of absolute error（MaxAE）

Maximum of absolute error（MaxAE）measures the maximum of absolute error of the predicted temperature field.

$${E_{MaxAE} } = \mathop {\max }\limits_{({x_i},{y_j}) \in \Omega } \left| {T({x_i},{y_j}) - T'({x_i},{y_j})} \right|$$

Component-constrained mean absolute error（CMAE）

Component-constrained mean absolute error（CMAE）computes the mean value of the absolute error over the heat-source component.

$${E_{CMAE} } = \frac{1}{ {\left| { {\Omega _c} } \right|} }\sum\limits_{({x_i},{y_j}) \in {\Omega _c} } {\left| {T({x_i},{y_j}) - T'({x_i},{y_j})} \right|}$$

Maximum of component-constrained absolute error（M-CAE）

Maximum of component-constrained absolute error（M-CAE）describes the maximum error of the predicted temperature field over the heat-source components.

$${E_{CMAE} } = \mathop {\max }\limits_{({x_i},{y_j}) \in {\Omega _c} } \left| {T({x_i},{y_j}) - T'({x_i},{y_j})} \right|$$

Boundary-constrained mean absolute error（BMAE）

Boundary-constrained mean absolute error（BMAE）computes the mean value of the absolute error near the boundaries of the heat-source systems.

$${E_{BMAE} } = \frac{1}{ {\left| { {\Omega _b} } \right|} }\sum\limits_{({x_i},{y_j}) \in {\Omega _b} } {\left| {T({x_i},{y_j}) - T'({x_i},{y_j})} \right|}$$