🐛[BUG]: Validator L2 Relative Error Loss

[Chawit K.]

Version

v1.3.0

On which installation method(s) does this occur?

No response

Describe the issue

I would like to report an issue in modulus/sym/domain/validator/validator.py.

As in lines 39 - 42, the L2 relative error is written as

$$ e_{L^2} = \sqrt{\frac{ \frac{1}{N} \sum_{i} \left( u_{true_i} - u_{pred_i} \right)^2} {\frac{1}{N-\delta N} \sum_i \left(u_{true_i} - \overline{u}_{true}\right)^2}} $$

new_var["l2_relative_error_" + str(key)] = torch.sqrt(
                torch.mean(torch.square(true_var[key] - pred_var[key]))
                / torch.var(true_var[key])
            )

However, I understand that the L2 relative error is defined as

$$ e_{L^2} = \sqrt{ \frac{ \sum_{i} \left( u_{true_i}-u_{pred_i} \right)^2} {\sum_i \left(u_{true_i}\right)^2} } $$

new_var["l2_relative_error_" + str(key)] = torch.sqrt(
                torch.sum(torch.square(true_var[key] - pred_var[key]))
                /  torch.sum(torch.square(true_var[key]))
            )

Regarding the validation loss, I would like to bring up this concern. If I misunderstand the concept of the function, please correct my understanding. Thank you very much for your consideration.

Minimum reproducible example

No response

Relevant log output

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Environment details

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Other/Misc.

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