Add physical context to examples, and use backend-agnostic functions by vishnu-2206 · Pull Request #2066 · lululxvi/deepxde

vishnu-2206 · 2026-03-19T09:14:43Z

As a physicist working with PINNs, I noticed that the 1D diffusion example lacked a physical description of the source term.

I have :

Added a detailed Module Docstring explaining the 1D Heat Equation context.
Refactored the pde function to clearly identify the forcing term $f(x, t)$.
Ensured that the source term is derived to satisfy the analytical solution $y = e^{-t} \sin(\pi x)$.

This makes the example more accessible for students and researchers using DeepXDE for scientific modeling.

…source term

echen5503 · 2026-03-19T13:41:43Z

-        * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
+        + source_term_val
    )
+


Minimal changes, don't change any spaces not explicitly part of PR

echen5503 · 2026-03-19T13:44:04Z

+∂y/∂t - ∂²y/∂x² = f(x, t)
+where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
+
+Physical context: This represents a 1D rod with a decaying heat source, where the ends are kept at zero temperature (Dirichlet Boundary Conditions).


No need for this, this doesn't exist in other examples

echen5503 · 2026-03-19T13:48:43Z

+
+    # Physics Note: The following term is the forced heat source f(x, t)
+    # required to satisfy the analytical solution y = e^(-t)sin(πx).
+    source_term_val = tf.exp(-x[:, 1:]) * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))


With this, note that you are making the tensorflow code different than the other backends. Also, note that this isn't the main point of the PR. If you really want to make this change, make it consistent for all backends, or use dde.backend.sin, dde.backend.exp, etc.

vishnu-2206 · 2026-03-19T15:06:48Z

Thank you for the guidance @echen5503. I have updated the PR to: 1. Revert the docstring and spacing to match the existing minimalist style of the library. 2. Implement the source term using dde.backend to ensure consistency and compatibility across all supported backends (TF, PyTorch, JAX, Paddle). Let me know if any other adjustments are required.

…

On Thu, 19 Mar 2026 at 19:19, Edwin Chen ***@***.***> wrote: ***@***.**** requested changes on this pull request. ------------------------------ In examples/pinn_forward/diffusion_1d.py <#2066 (comment)>: > ) + Minimal changes, don't change any spaces not explicitly part of PR ------------------------------ In examples/pinn_forward/diffusion_1d.py <#2066 (comment)>: > @@ -1,4 +1,14 @@ -"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle""" +""" +Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle +1D Diffusion Equation with a Time-Dependent Source Term. + +This example solves the heat equation: +∂y/∂t - ∂²y/∂x² = f(x, t) +where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx). + +Physical context: This represents a 1D rod with a decaying heat source, where the ends are kept at zero temperature (Dirichlet Boundary Conditions). No need for this, this doesn't exist in other examples ------------------------------ In examples/pinn_forward/diffusion_1d.py <#2066 (comment)>: > @@ -18,27 +28,34 @@ def pde(x, y): # Backend jax # dy_t, _ = dde.grad.jacobian(y, x, j=1) # dy_xx, _ = dde.grad.hessian(y, x, j=0) - # Backend tensorflow.compat.v1 or tensorflow + + # Physics Note: The following term is the forced heat source f(x, t) + # required to satisfy the analytical solution y = e^(-t)sin(πx). + source_term_val = tf.exp(-x[:, 1:]) * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1])) With this, note that you are making the tensorflow code different than the other backends. Also, note that this isn't the main point of the PR. If you really want to make this change, make it consistent for all backends, or use dde.backend.sin, dde.backend.exp, etc. — Reply to this email directly, view it on GitHub <#2066 (review)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/BPVR7W22GDEMIXVNK5V6QID4RP3FRAVCNFSM6AAAAACWXVW2JGVHI2DSMVQWIX3LMV43YUDVNRWFEZLROVSXG5CSMV3GSZLXHMZTSNZVGIYTIMBQHA> . You are receiving this because you authored the thread.Message ID: ***@***.***>

echen5503 · 2026-03-19T15:28:08Z

-∂y/∂t - ∂²y/∂x² = f(x, t)
-where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
-
-Physical context: This represents a 1D rod with a decaying heat source, where the ends are kept at zero temperature (Dirichlet Boundary Conditions).


I meant just remove L9, not remove the whole thing. The rest is useful.

echen5503 · 2026-03-19T15:39:45Z

@vishnu-2206 it should look like this

"""
Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle
1D Diffusion Equation with a Time-Dependent Source Term.

This example solves the heat equation:
∂y/∂t - ∂²y/∂x² = f(x, t)
where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
"""
import deepxde as dde
import numpy as np

def pde(x, y):
    # Most backends
    dy_t = dde.grad.jacobian(y, x, j=1)
    dy_xx = dde.grad.hessian(y, x, j=0)
    
    # Physics Note: The following term is the forced heat source f(x, t)
    # required to satisfy the analytical solution y = e^(-t)sin(πx).
    source_term_val = dde.backend.exp(-x[:, 1:]) * (dde.backend.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * dde.backend.sin(np.pi * x[:, 0:1]))

    return ( 
        dy_t
        - dy_xx
        + source_term_val
    ) 

def func(x):
    return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])


geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

bc = dde.icbc.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
ic = dde.icbc.IC(geomtime, func, lambda _, on_initial: on_initial)
data = dde.data.TimePDE(
    geomtime,
    pde,
    [bc, ic],
    num_domain=40,
    num_boundary=20,
    num_initial=10,
    solution=func,
    num_test=10000,
)

layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.FNN(layer_size, activation, initializer)

model = dde.Model(data, net)

model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(iterations=10000)

dde.saveplot(losshistory, train_state, issave=True, isplot=True)

echen5503 · 2026-03-19T15:41:26Z

By only using deepXDE functions we can make examples backend-agnostic without casework.

…correction

echen5503 · 2026-03-19T16:57:25Z

It seems you missed some of my comments in the previous iterations. The point of the PR is to add documentation and keep the code concise with dde.backend functions. However, I requested you to add back the documentation and you did not do so.

Secondly, please focus on one file at a time. We can fix others in future PRs. Changing more and more files makes the PR hard to review.

Finally, please make sure to clean your AI output. AI will write comments to denote where things have changed, but it is not necessary here.

To reiterate, your code should look like this:

"""
Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle
1D Diffusion Equation with a Time-Dependent Source Term.

This example solves the heat equation:
∂y/∂t - ∂²y/∂x² = f(x, t)
where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
"""
import deepxde as dde
import numpy as np

def pde(x, y):
    # Most backends
    dy_t = dde.grad.jacobian(y, x, j=1)
    dy_xx = dde.grad.hessian(y, x, j=0)
    
    # Physics Note: The following term is the forced heat source f(x, t)
    # required to satisfy the analytical solution y = e^(-t)sin(πx).
    source_term_val = dde.backend.exp(-x[:, 1:]) * (dde.backend.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * dde.backend.sin(np.pi * x[:, 0:1]))

    return ( 
        dy_t
        - dy_xx
        + source_term_val
    ) 

def func(x):
    return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])


geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

bc = dde.icbc.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
ic = dde.icbc.IC(geomtime, func, lambda _, on_initial: on_initial)
data = dde.data.TimePDE(
    geomtime,
    pde,
    [bc, ic],
    num_domain=40,
    num_boundary=20,
    num_initial=10,
    solution=func,
    num_test=10000,
)

layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.FNN(layer_size, activation, initializer)

model = dde.Model(data, net)

model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(iterations=10000)

dde.saveplot(losshistory, train_state, issave=True, isplot=True)

In summary, please follow the format which I have described, and only change 1 file. Once this is merged, feel free to open a PR for the other 2.

echen5503 · 2026-03-19T16:59:12Z

You can just copy-paste the code I provided verbatim. In future PRs, please keep my comments in mind.

…ackend

vishnu-2206 · 2026-03-19T17:04:58Z

Thank you for the guidance and the template. I have updated diffusion_1d.py to use source_term_val and dde.backend to ensure it is fully backend-agnostic as suggested. Once you've had a chance to verify this structure, I will apply the exact same logic to the other 1D diffusion examples (exactBC and resample) for consistency. Ready for review!

…

On Thu, 19 Mar 2026 at 22:29, Edwin Chen ***@***.***> wrote: *echen5503* left a comment (lululxvi/deepxde#2066) <#2066 (comment)> You can just copy-paste the code I provided verbatim. In future PRs, please keep my comments in mind. — Reply to this email directly, view it on GitHub <#2066 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/BPVR7W7AKY3WESJLVMGHJ6T4RQRPNAVCNFSM6AAAAACWXVW2JGVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHM2DAOJRGY4TSMZXHA> . You are receiving this because you were mentioned.Message ID: ***@***.***>

echen5503 · 2026-03-19T17:07:00Z

Thank you, this looks good. You can modify the other files to have the same structure.

vishnu-2206 · 2026-03-19T17:09:05Z

Thank you for the feedback. Will modify the rest as well.

…

On Thu, 19 Mar 2026 at 22:37, Edwin Chen ***@***.***> wrote: *echen5503* left a comment (lululxvi/deepxde#2066) <#2066 (comment)> Thank you, this looks good. You can modify the other files to have the same structure. — Reply to this email directly, view it on GitHub <#2066 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/BPVR7W5YJ4SREFUMWZNASXD4RQSMVAVCNFSM6AAAAACWXVW2JGVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHM2DAOJRG43TMNRSHA> . You are receiving this because you were mentioned.Message ID: ***@***.***>

…tructure

echen5503 · 2026-03-20T19:21:43Z

LGTM. Care to add physics docs to 1d_resample as well?

echen5503 · 2026-03-20T22:20:20Z

Once added, I think this will be ready to merge. Please update the title of the PR accordingly, to something like "Add physical context to examples, and use backend-agnostic functions".

vishnu-2206 · 2026-03-21T09:54:44Z

I have updated the title and applied the same logic and structure to diffusion_1d_resample.py for consistency across the suite. These updates ensure physical accuracy and full compatibility with all DeepXDE backends by using dde.backend functions.

echen5503

Just a small nitpick then we can merge this. Thanks for your help!

echen5503 · 2026-03-19T16:46:34Z

+    f = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
+    # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle


Suggested change

# Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle

No need to say this, because it is implied

echen5503 · 2026-03-19T16:47:55Z

+    f = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
+    # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle


Suggested change

# Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle

echen5503 · 2026-03-19T16:48:29Z

+    f = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
+    # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle


Suggested change

# Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle

echen5503 · 2026-03-22T22:38:28Z

-    # lambda x, y: x[..., 1:2] * (1 - x[..., 0:1] ** 2) * y + jnp.sin(np.pi * x[..., 0:1])
-    # Backend paddle
-    # lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + paddle.sin(np.pi * x[:, 0:1])
+    # This works for TensorFlow, PyTorch, JAX, and Paddle


No need to say this.

vishnu-2206 · 2026-03-23T07:22:42Z

Thank you for the review! I've removed the redundant backend comments and ensured a cleaner structure across diffusion_1d.py, exactBC.py, and resample.py. Ready for the final merge!

…

On Mon, 23 Mar 2026 at 04:49, Edwin Chen ***@***.***> wrote: ***@***.**** requested changes on this pull request. Just a small nitpick then we can merge this. Thanks for your help! ------------------------------ In examples/pinn_forward/diffusion_1d.py <#2066 (comment)>: > @@ -18,34 +18,16 @@ def pde(x, y): # Backend jax # dy_t, _ = dde.grad.jacobian(y, x, j=1) # dy_xx, _ = dde.grad.hessian(y, x, j=0) - # Backend tensorflow.compat.v1 or tensorflow + # Cross-backend source term + f = dde.backend.exp(-x[:, 1:]) * ( + dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1]) + ) + # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ⬇️ Suggested change - # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ------------------------------ In examples/pinn_forward/diffusion_1d_exactBC.py <#2066 (comment)>: > @@ -18,34 +18,16 @@ def pde(x, y): # Backend jax # dy_t, _ = dde.grad.jacobian(y, x, i=0, j=1) # dy_xx, _ = dde.grad.hessian(y, x, i=0, j=0) - # Backend tensorflow.compat.v1 or tensorflow + # Cross-backend source term + f = dde.backend.exp(-x[:, 1:]) * ( + dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1]) + ) + # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ⬇️ Suggested change - # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ------------------------------ In examples/pinn_forward/diffusion_1d_resample.py <#2066 (comment)>: > @@ -18,36 +18,18 @@ def pde(x, y): # Backend jax # dy_t, _ = dde.grad.jacobian(y, x, i=0, j=1) # dy_xx, _ = dde.grad.hessian(y, x, i=0, j=0) - # Backend tensorflow.compat.v1 or tensorflow + # Cross-backend source term + f = dde.backend.exp(-x[:, 1:]) * ( + dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1]) + ) + # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ⬇️ Suggested change - # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle ------------------------------ In examples/pinn_forward/diffusion_1d_exactBC.py <#2066 (comment)>: > @@ -63,14 +50,8 @@ def func(x): initializer = "Glorot uniform" net = dde.nn.FNN(layer_size, activation, initializer) net.apply_output_transform( - # Backend tensorflow.compat.v1 or tensorflow - lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + tf.sin(np.pi * x[:, 0:1]) - # Backend pytorch - # lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + torch.sin(np.pi * x[:, 0:1]) - # Backend jax - # lambda x, y: x[..., 1:2] * (1 - x[..., 0:1] ** 2) * y + jnp.sin(np.pi * x[..., 0:1]) - # Backend paddle - # lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + paddle.sin(np.pi * x[:, 0:1]) + # This works for TensorFlow, PyTorch, JAX, and Paddle No need to say this. ------------------------------ In examples/pinn_forward/diffusion_1d.py <#2066 (comment)>: > @@ -18,34 +18,16 @@ def pde(x, y): # Backend jax # dy_t, _ = dde.grad.jacobian(y, x, j=1) # dy_xx, _ = dde.grad.hessian(y, x, j=0) - # Backend tensorflow.compat.v1 or tensorflow + # Cross-backend source term + f = dde.backend.exp(-x[:, 1:]) * ( + dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1]) + ) + # Backend tensorflow.compat.v1 or tensorflow, pytorch, jax, paddle No need to say this, because it is implied — Reply to this email directly, view it on GitHub <#2066?email_source=notifications&email_token=BPVR7W2I6OZYNTKSRBWBGIT4SBYIHA5CNFSNUABKM5UWIORPF5TWS5BNNB2WEL2QOVWGYUTFOF2WK43UKJSXM2LFO4XTGOJXGY2TANZTGQ22M4TFMFZW63VHNVSW45DJN5XKKZLWMVXHJLDGN5XXIZLSL5RWY2LDNM#pullrequestreview-3976507345>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/BPVR7W6AUMGGCSUFREFC67T4SBYIHAVCNFSM6AAAAACWXVW2JGVHI2DSMVQWIX3LMV43YUDVNRWFEZLROVSXG5CSMV3GSZLXHMZTSNZWGUYDOMZUGU> . You are receiving this because you were mentioned.Message ID: ***@***.***>

echen5503 · 2026-03-23T16:50:17Z

@lululxvi looks ready for merge.

Please credit @vishnu2206 (first), and @echen5503 (second).

vishnu-2206 · 2026-03-24T08:02:59Z

Thank you! Please credit @vishnu2206 as the primary author and @echen5503 for the guidance and review. I appreciate the help in getting these examples to the library standard!

vishnu-2206 · 2026-03-31T18:08:39Z

I have amended the commit to include the co-author trailers as requested, crediting @vishnu-2206 (primary) and @echen5503 (secondary). The metadata should now be correctly reflected in the PR. Ready for final review.

echen5503 · 2026-04-01T16:49:15Z

Hi, suggested code changes look good. But who is this Carrot-Bear account?

Co-authored-by: vishnu-2206 <48398436+vishnu-2206@users.noreply.github.com> Co-authored-by: echen5503 <87834510+echen5503@users.noreply.github.com>

vishnu-2206 · 2026-04-03T05:11:32Z

I have corrected my local Git configuration and amended the commit. The placeholder 'Carrot-Bear' has been replaced with my professional name, and the co-author metadata has been verified. The PR is now ready for your final review and merge.

echen5503 · 2026-04-03T05:18:24Z

Looks good.

Refractor diffusion_1d example with physical context and descriptive …

1508cad

…source term

echen5503 suggested changes Mar 19, 2026

View reviewed changes

chore: use dde.backend for source term and revert docstring changes

64a176f

echen5503 suggested changes Mar 19, 2026

View reviewed changes

fix: unify 1D diffusion suite with universal dde.backend and physics …

f1fbe3a

…correction

fix: refactor 1D diffusion example to be backend-agnostic using dde.b…

f355cb5

…ackend

fix: update exactBC.py with backend-agnostic transform and approved s…

f0d08ae

…tructure

add 1D heat equation example with adaptive resampling

5321b8f

vishnu-2206 changed the title ~~doc: add physical context and clarify 1D diffusion source term~~ Add physical context to examples, and use backend-agnostic functions Mar 21, 2026

echen5503 suggested changes Mar 22, 2026

View reviewed changes

vishnu-2206 requested review from echen5503 March 31, 2026 17:21

vishnu-2206 force-pushed the doc-physics-diffusion-vsr branch from 6ca275d to 0ea5d74 Compare March 31, 2026 18:07

Remove redundant backend labels and cleanup comments

b2f54be

Co-authored-by: vishnu-2206 <48398436+vishnu-2206@users.noreply.github.com> Co-authored-by: echen5503 <87834510+echen5503@users.noreply.github.com>

vishnu-2206 force-pushed the doc-physics-diffusion-vsr branch from 0ea5d74 to b2f54be Compare April 3, 2026 05:11

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