Stabilize DPM++, especially for SDXL and SDE-DPM++ (#5541)

* stabilize dpmpp for sdxl by using euler at the final step

* add lu's uniform logsnr time steps

* add test

* fix check_copies

* fix tests

---------
Co-authored-by: Patrick von Platen <patrick.v.platen@gmail.com>

src/diffusers/schedulers/scheduling_dpmsolver_multistep.py

@@ -117,9 +117,17 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         lower_order_final (`bool`, defaults to `True`):
             Whether to use lower-order solvers in the final steps. Only valid for < 15 inference steps. This can
             stabilize the sampling of DPMSolver for steps < 15, especially for steps <= 10.
+        euler_at_final (`bool`, defaults to `False`):
+            Whether to use Euler's method in the final step. It is a trade-off between numerical stability and detail
+            richness. This can stabilize the sampling of the SDE variant of DPMSolver for small number of inference
+            steps, but sometimes may result in blurring.
         use_karras_sigmas (`bool`, *optional*, defaults to `False`):
             Whether to use Karras sigmas for step sizes in the noise schedule during the sampling process. If `True`,
             the sigmas are determined according to a sequence of noise levels {σi}.
+        use_lu_lambdas (`bool`, *optional*, defaults to `False`):
+            Whether to use the uniform-logSNR for step sizes proposed by Lu's DPM-Solver in the noise schedule during
+            the sampling process. If `True`, the sigmas and time steps are determined according to a sequence of
+            `lambda(t)`.
         lambda_min_clipped (`float`, defaults to `-inf`):
             Clipping threshold for the minimum value of `lambda(t)` for numerical stability. This is critical for the
             cosine (`squaredcos_cap_v2`) noise schedule.
@@ -154,7 +162,9 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         algorithm_type: str = "dpmsolver++",
         solver_type: str = "midpoint",
         lower_order_final: bool = True,
+        euler_at_final: bool = False,
         use_karras_sigmas: Optional[bool] = False,
+        use_lu_lambdas: Optional[bool] = False,
         lambda_min_clipped: float = -float("inf"),
         variance_type: Optional[str] = None,
         timestep_spacing: str = "linspace",
@@ -258,6 +268,12 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
             sigmas = self._convert_to_karras(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
             timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas]).round()
             sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
+        elif self.config.use_lu_lambdas:
+            lambdas = np.flip(log_sigmas.copy())
+            lambdas = self._convert_to_lu(in_lambdas=lambdas, num_inference_steps=num_inference_steps)
+            sigmas = np.exp(lambdas)
+            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas]).round()
+            sigmas = np.concatenate([sigmas, sigmas[-1:]]).astype(np.float32)
         else:
             sigmas = np.interp(timesteps, np.arange(0, len(sigmas)), sigmas)
             sigma_last = ((1 - self.alphas_cumprod[0]) / self.alphas_cumprod[0]) ** 0.5
@@ -354,6 +370,19 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         sigmas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
         return sigmas
 
+    def _convert_to_lu(self, in_lambdas: torch.FloatTensor, num_inference_steps) -> torch.FloatTensor:
+        """Constructs the noise schedule of Lu et al. (2022)."""
+
+        lambda_min: float = in_lambdas[-1].item()
+        lambda_max: float = in_lambdas[0].item()
+
+        rho = 1.0  # 1.0 is the value used in the paper
+        ramp = np.linspace(0, 1, num_inference_steps)
+        min_inv_rho = lambda_min ** (1 / rho)
+        max_inv_rho = lambda_max ** (1 / rho)
+        lambdas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
+        return lambdas
+
     def convert_model_output(
         self,
         model_output: torch.FloatTensor,
@@ -787,8 +816,9 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         if self.step_index is None:
             self._init_step_index(timestep)
 
-        lower_order_final = (
-            (self.step_index == len(self.timesteps) - 1) and self.config.lower_order_final and len(self.timesteps) < 15
+        # Improve numerical stability for small number of steps
+        lower_order_final = (self.step_index == len(self.timesteps) - 1) and (
+            self.config.euler_at_final or (self.config.lower_order_final and len(self.timesteps) < 15)
         )
         lower_order_second = (
             (self.step_index == len(self.timesteps) - 2) and self.config.lower_order_final and len(self.timesteps) < 15

src/diffusers/schedulers/scheduling_dpmsolver_multistep_inverse.py

@@ -117,6 +117,10 @@ class DPMSolverMultistepInverseScheduler(SchedulerMixin, ConfigMixin):
         lower_order_final (`bool`, defaults to `True`):
             Whether to use lower-order solvers in the final steps. Only valid for < 15 inference steps. This can
             stabilize the sampling of DPMSolver for steps < 15, especially for steps <= 10.
+        euler_at_final (`bool`, defaults to `False`):
+            Whether to use Euler's method in the final step. It is a trade-off between numerical stability and detail
+            richness. This can stabilize the sampling of the SDE variant of DPMSolver for small number of inference
+            steps, but sometimes may result in blurring.
         use_karras_sigmas (`bool`, *optional*, defaults to `False`):
             Whether to use Karras sigmas for step sizes in the noise schedule during the sampling process. If `True`,
             the sigmas are determined according to a sequence of noise levels {σi}.
@@ -154,6 +158,7 @@ class DPMSolverMultistepInverseScheduler(SchedulerMixin, ConfigMixin):
         algorithm_type: str = "dpmsolver++",
         solver_type: str = "midpoint",
         lower_order_final: bool = True,
+        euler_at_final: bool = False,
         use_karras_sigmas: Optional[bool] = False,
         lambda_min_clipped: float = -float("inf"),
         variance_type: Optional[str] = None,
@@ -804,8 +809,9 @@ class DPMSolverMultistepInverseScheduler(SchedulerMixin, ConfigMixin):
         if self.step_index is None:
             self._init_step_index(timestep)
 
-        lower_order_final = (
-            (self.step_index == len(self.timesteps) - 1) and self.config.lower_order_final and len(self.timesteps) < 15
+        # Improve numerical stability for small number of steps
+        lower_order_final = (self.step_index == len(self.timesteps) - 1) and (
+            self.config.euler_at_final or (self.config.lower_order_final and len(self.timesteps) < 15)
         )
         lower_order_second = (
             (self.step_index == len(self.timesteps) - 2) and self.config.lower_order_final and len(self.timesteps) < 15

tests/schedulers/test_scheduler_dpm_multi.py

@@ -29,6 +29,7 @@ class DPMSolverMultistepSchedulerTest(SchedulerCommonTest):
             "algorithm_type": "dpmsolver++",
             "solver_type": "midpoint",
             "lower_order_final": False,
+            "euler_at_final": False,
             "lambda_min_clipped": -float("inf"),
             "variance_type": None,
         }
@@ -195,6 +196,10 @@ class DPMSolverMultistepSchedulerTest(SchedulerCommonTest):
         self.check_over_configs(lower_order_final=True)
         self.check_over_configs(lower_order_final=False)
 
+    def test_euler_at_final(self):
+        self.check_over_configs(euler_at_final=True)
+        self.check_over_configs(euler_at_final=False)
+
     def test_lambda_min_clipped(self):
         self.check_over_configs(lambda_min_clipped=-float("inf"))
         self.check_over_configs(lambda_min_clipped=-5.1)
@@ -258,6 +263,12 @@ class DPMSolverMultistepSchedulerTest(SchedulerCommonTest):
 
         assert abs(result_mean.item() - 0.2096) < 1e-3
 
+    def test_full_loop_with_lu_and_v_prediction(self):
+        sample = self.full_loop(prediction_type="v_prediction", use_lu_lambdas=True)
+        result_mean = torch.mean(torch.abs(sample))
+
+        assert abs(result_mean.item() - 0.1554) < 1e-3
+
     def test_switch(self):
         # make sure that iterating over schedulers with same config names gives same results
         # for defaults