# EulerSolver

### *class* deepinv.sampling.EulerSolver(timesteps=None, t_start=None, t_end=None, num_steps=None, rng=None)

Bases: [`BaseSDESolver`](https://deepinv.org/api/stubs/deepinv.sampling.BaseSDESolver.html.md#deepinv.sampling.BaseSDESolver)

Euler-Maruyama solver for SDEs.

This solver uses the Euler-Maruyama method to numerically integrate SDEs. It is a first-order method that
approximates the solution using the following update rule:

$$
x_{t+dt} = x_t + f(x_t,t)dt + g(t) W_{dt}
$$

where $W_t$ is a Gaussian random variable with mean 0 and variance dt.

* **Parameters:**
  * **timesteps** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* [*numpy.ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray) *,* [*list*](https://docs.python.org/3.12/library/stdtypes.html#list)) – The time steps at which to evaluate the solution.
  * **timesteps** – time steps at which the SDE will be discretized.
  * **t_start** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – the starting time of the SDE, optional. If not provided, it will be inferred from the `timesteps` argument.
  * **t_end** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – the ending time of the SDE, optional. If not provided, it will be inferred from the `timesteps` argument.
  * **num_steps** ([*int*](https://docs.python.org/3.12/library/functions.html#int)) – the number of time steps for the SDE, optional. If not provided, it will be inferred from the `timesteps` argument.
  * **rng** ([*torch.Generator*](https://docs.pytorch.org/docs/stable/generated/torch.Generator.html#torch.Generator)) – A random number generator for reproducibility.

#### NOTE
You can either provide the `timesteps` argument directly, or specify `t_start`, `t_end`, and `num_steps` to generate the time steps automatically (linearly with constant stepsize). If both are provided, the `timesteps` argument will take precedence.

#### step(sde, t0, t1, x0, \*args, \*\*kwargs)

Perform a single Euler-Maruyama step from time `t0` to time `t1`, with current state `x0`.

* **Parameters:**
  * **sde** ([*deepinv.sampling.BaseSDE*](https://deepinv.org/api/stubs/deepinv.sampling.BaseSDE.html.md#deepinv.sampling.BaseSDE)) – the SDE to solve.
  * **t0** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *or* [*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Time at the start of the step, of size (,).
  * **t1** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *or* [*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Time at the end of the step, of size (,).
  * **x0** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Current state of the system, of size (batch_size, d).
  * **\*args** – additional arguments for the drift of the SDE.
  * **\*\*kwargs** – additional keyword arguments for the drift of the SDE.
* **Return torch.Tensor, int:**
  Updated state of the system after the step and number of function evaluations (NFE) performed during the step (here 1).
* **Return type:**
  [tuple](https://docs.python.org/3.12/library/stdtypes.html#tuple)[[torch.Tensor](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor), [int](https://docs.python.org/3.12/library/functions.html#int)]

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## Examples using `EulerSolver`:

<div class="sphx-glr-thumbnails">
<!-- thumbnail-parent-div-open --><div class="sphx-glr-thumbcontainer" tooltip="This demo shows you how to use deepinv.sampling.PosteriorDiffusion to perform posterior sampling. It also can be used to perform unconditional image generation with arbitrary denoisers, if the data fidelity term is not specified.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_diffusion_sde_thumb.png)

[Building your diffusion posterior sampling method using SDEs](https://deepinv.org/auto_examples/sampling/demo_diffusion_sde.html.md)

  <div class="sphx-glr-thumbnail-title">Building your diffusion posterior sampling method using SDEs</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This demo shows you how to perform unconditional image generation and posterior sampling using Flow Matching (FM).">![](auto_examples/sampling/images/thumb/sphx_glr_demo_flow_matching_thumb.png)

[Flow-Matching for posterior sampling and unconditional generation](https://deepinv.org/auto_examples/sampling/demo_flow_matching.html.md)

  <div class="sphx-glr-thumbnail-title">Flow-Matching for posterior sampling and unconditional generation</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example compares six approximations of the measurement-matching term used by diffusion posterior samplers.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_noisy_data_fidelity_thumb.png)

[Noisy data-fidelity terms for diffusion posterior sampling](https://deepinv.org/auto_examples/sampling/demo_noisy_data_fidelity.html.md)

  <div class="sphx-glr-thumbnail-title">Noisy data-fidelity terms for diffusion posterior sampling</div>
</div>
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