Init commit

.gitignore

+__pycache__/
+*.py[cod]
+*$py.class
+explore*
+*data/
+*backup/
+*vdb/
+build/
+*.npy
+*.csv
+.vscode
+.venv*
+*.dll
\ No newline at end of file

LICENCE

+MIT License
+
+Copyright © 2021, ETH Zurich, Jingwei Tang
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.

README.md

+# Shrunk-Domain
+
+Official PyTorch implementation for [Honey, I Shrunk the Domain: Frequency-aware Force Field Reduction for Efficient Fluids Optimization](https://cgl.ethz.ch/publications/papers/paperTan21a.php).
+
+### Installation
+``` bash
+# create a conda env
+conda create -n shrunk-domain
+conda activate shrunk-domain
+# install pytorch
+conda install pytorch==1.7.1 torchvision==0.8.2 cudatoolkit=11.0 -c pytorch
+# install other requirements
+pip install -r requirements.txt
+```
+
+### Simulation
+The sequences for the benchmark tests can be generated through simulation by running the provided simulation scripts. For example, to generate 2D uniform-wind simulation:
+```bash
+bash scripts/simulate_2d_uniform.sh
+```
+
+### Optimization
+Optimizations can be used for keyframe matching. Before running the optimization scripts, we need to put the keyframe data in `.npz` format into `simulations/data` directory. For example, the above 2D 128x128 uniform-wind simulation will create `.npz` files under `simulations/data/benchmark/2d_128_uniform`. A corresponding `opt.txt` file is also needed to specify simulation configurations as guidance to the simulations used in the optimization process.
+
+After the data and config files are prepared, we can start to run the optimization. The provided scripts give examples to run different optimization methods: baseline+TV `bash scripts/optimize_2d_baseline.sh`, progressive upsampling `bash scripts/optimize_2d_prog_force.sh` and ours `bash scripts/optimize_2d_prog_freq.sh`.
+
+After the optimizaiton is done, we can run a simulation by loading the saved optimized parameter to get to the target frame:
+
+``` bash
+# need to specify EXPERIMENT_NAME in this script first
+bash scripts/optimize_simulate.sh
+```
+
+### Structure of the code
+- `lib/dfluids` implements differentiable operations needed to run smoke simulations.
+- `lib/PyTorch_LBFGS.py` is adapted from [PyTorch-LBFGS](https://github.com/hjmshi/PyTorch-LBFGS), the optimization processes use this implementation of LBFGS optimizer.
+- `simulations` defines argparse interfaces for users to define simulation config, and uses operations from `dfluids` to run smoke simulations.
+- `optimizations/models` defines models to wrap optimization parameters and how the parameters are used in the simulation.
+- `optimizations/optimizations` specifies optimization processes, e.g. procedures to compute objective function, to optimize the parameters, etc.
+- `forces` contains scripts to generate force fields used for the benchmark tests.

forces/generate_multi_vortex_128.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+# 2d big curl force
+res = 128
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+curl = 0
+
+# large scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.0]),
+    sig=torch.Tensor([[0.1, 0], [0, 0.1]]),
+    amp=torch.Tensor([-0.02]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# middle scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.4, -0.4]),
+    sig=torch.Tensor([[0.04, 0], [0, 0.04]]),
+    amp=torch.Tensor([0.015]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.35, 0.1]),
+    sig=torch.Tensor([[0.01, 0], [0, 0.01]]),
+    amp=torch.Tensor([0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# small scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.2, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.006]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.25, 0.4]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.006]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.5, 0.35]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.006]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.006]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.0]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.006]),
+)
+curl = curl + node_op.get_curl(phi)
+
+utils.mkdir("forces/data/2d_128_multi_vortex/")
+mac_op.save_img(curl, "forces/data/2d_128_multi_vortex/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_128_multi_vortex/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_128_multi_vortex/000.npz", force=mac_op.grid2array(curl))

forces/generate_multi_vortex_256.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+# 2d big curl force
+res = 256
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+curl = 0
+
+# large scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.0]),
+    sig=torch.Tensor([[0.1, 0], [0, 0.1]]),
+    amp=torch.Tensor([-0.03]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# middle scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.4, -0.4]),
+    sig=torch.Tensor([[0.04, 0], [0, 0.04]]),
+    amp=torch.Tensor([0.03]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.35, 0.1]),
+    sig=torch.Tensor([[0.01, 0], [0, 0.01]]),
+    amp=torch.Tensor([0.02]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# small scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.2, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.25, 0.4]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.5, 0.35]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.0]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+
+utils.mkdir("forces/data/2d_256_multi_vortex/")
+mac_op.save_img(curl, "forces/data/2d_256_multi_vortex/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_256_multi_vortex/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_256_multi_vortex/000.npz", force=mac_op.grid2array(curl))

forces/generate_multi_vortex_64.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+# 2d big curl force
+res = 64
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+curl = 0
+
+# large scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.0]),
+    sig=torch.Tensor([[0.1, 0], [0, 0.1]]),
+    amp=torch.Tensor([-0.015]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# middle scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.4, -0.4]),
+    sig=torch.Tensor([[0.04, 0], [0, 0.04]]),
+    amp=torch.Tensor([0.01]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.35, 0.1]),
+    sig=torch.Tensor([[0.01, 0], [0, 0.01]]),
+    amp=torch.Tensor([0.005]),
+)
+curl = curl + node_op.get_curl(phi)
+
+# small scale
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.2, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.002]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.25, 0.4]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.002]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.5, 0.35]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.002]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.2]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([0.002]),
+)
+curl = curl + node_op.get_curl(phi)
+
+phi = node_op.get_gaussian(
+    mu=torch.Tensor([-0.7, 0.0]),
+    sig=torch.Tensor([[0.001, 0], [0, 0.001]]),
+    amp=torch.Tensor([-0.002]),
+)
+curl = curl + node_op.get_curl(phi)
+
+
+utils.mkdir("forces/data/2d_64_multi_vortex/")
+mac_op.save_img(curl, "forces/data/2d_64_multi_vortex/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_64_multi_vortex/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_64_multi_vortex/000.npz", force=mac_op.grid2array(curl))

forces/generate_s_shape_128.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+# 2d big curl force
+res = 128
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+phi1 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, -0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([-0.08]),
+)
+
+phi2 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([0.08]),
+)
+
+curl1 = node_op.get_curl(phi1)
+curl2 = node_op.get_curl(phi2)
+
+curl = curl1
+curl[:, int(res / 2) :, :] = curl2[:, int(res / 2) :, :]
+
+
+utils.mkdir("forces/data/2d_128_s_shape/")
+mac_op.save_img(curl, "forces/data/2d_128_s_shape/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_128_s_shape/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_128_s_shape/000.npz", force=mac_op.grid2array(curl))

forces/generate_s_shape_256.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+
+res = 256
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+phi1 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, -0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([-0.15]),
+)
+
+phi2 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([0.15]),
+)
+
+curl1 = node_op.get_curl(phi1)
+curl2 = node_op.get_curl(phi2)
+
+curl = curl1
+curl[:, int(res / 2) :, :] = curl2[:, int(res / 2) :, :]
+
+
+utils.mkdir("forces/data/2d_256_s_shape/")
+mac_op.save_img(curl, "forces/data/2d_256_s_shape/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_256_s_shape/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_256_s_shape/000.npz", force=mac_op.grid2array(curl))

forces/generate_s_shape_64.py

+import os
+import sys
+import torch
+import numpy as np
+
+sys.path.append(os.getcwd())
+from lib.dfluids.solver import FluidSolver
+from lib.dfluids.grid import GridType
+import lib.dfluids.utils as utils
+
+# 2d big curl force
+res = 64
+solver = FluidSolver(grid_size=(res, res))
+ext = solver.create_extforces()
+mac_op = solver.create_grid_operator(GridType.MAC)
+real_op = solver.create_grid_operator(GridType.REAL)
+node_op = solver.create_grid_operator(GridType.NODE)
+flags, phi_obs = ext.init_flags(phi=True, boundary_width=0, wall="xXyY")
+
+phi1 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, -0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([-0.04]),
+)
+
+phi2 = node_op.get_gaussian(
+    mu=torch.Tensor([0.0, 0.5]),
+    sig=torch.Tensor([[40, 0], [0, 1]]),
+    amp=torch.Tensor([0.04]),
+)
+
+curl1 = node_op.get_curl(phi1)
+curl2 = node_op.get_curl(phi2)
+
+curl = curl1
+curl[:, int(res / 2) :, :] = curl2[:, int(res / 2) :, :]
+
+utils.mkdir("forces/data/2d_64_s_shape/")
+mac_op.save_img(curl, "forces/data/2d_64_s_shape/lic.png", colormap="LIC")
+mac_op.save_img(curl, "forces/data/2d_64_s_shape/arrow.png", colormap="ARROW")
+np.savez("forces/data/2d_64_s_shape/000.npz", force=mac_op.grid2array(curl))

lib/dfluids/advection.py

+import torch
+import torch.nn.functional as F
+
+from .grid import GridType
+from . import utils
+
+
+class Advection(object):
+    def __init__(self, solver):
+        self.solver = solver
+        self.device = self.solver.get_device()
+        self.mac_op = self.solver.create_grid_operator(GridType.MAC)
+        self.real_op = self.solver.create_grid_operator(GridType.REAL)
+        self.node_op = self.solver.create_grid_operator(GridType.NODE)
+        self.ext = self.solver.create_extforces()
+
+    def advect_semi_lagrange(
+        self, flags, phi_obs, vel, rho, dt=None, order=1, clamp_mode=2, rk_order=3
+    ):
+        if dt is None:
+            dt = self.solver.get_dt()
+        if order == 1:
+            rho_adv = self._advect_semi_lagrange(vel, rho, dt, rk_order)
+        elif order == 2:
+            rho_fwd = self._advect_semi_lagrange(vel, rho, dt, rk_order)
+            rho_bwd = self._advect_semi_lagrange(vel, rho_fwd, -dt, rk_order)
+            rho_corr = rho_fwd + (rho - rho_bwd) * 0.5
+            rho_adv = self._maccormack_clamp(
+                vel, rho_corr, rho_fwd, rho, dt, clamp_mode, rk_order
+            )
+        else:
+            raise ValueError("Advection order must be 1 or 2.")
+        rho_adv = self.ext.set_wall_bcs(phi_obs, rho_adv)
+        return rho_adv
+
+    def _advect_semi_lagrange(self, vel, rho, dt, rk_order):
+        if rho.size() == self.solver.get_torch_size_real():
+            return self._advect_real_semi_lagrange(vel, rho, dt, rk_order)
+        elif rho.size() == self.solver.get_torch_size_mac():
+            return self._advect_mac_semi_lagrange(vel, rho, dt, rk_order)
+        # WARNING: does not distinguish 3D MAC and 3D NODE
+        elif rho.size() == self.solver.get_torch_size_node():
+            return self._advect_node_semi_lagrange(vel, rho, dt, rk_order)
+        else:
+            raise ValueError("rho has a wrong size: {}".format(rho.size()))
+
+    def _advect_real_semi_lagrange(self, vel, rho, dt, rk_order):
+        return self._advect_semi_lagrange_single_channel(
+            vel=vel, rho=rho, dt=dt, rk_order=rk_order, sample_pos="center"
+        )
+
+    def _advect_node_semi_lagrange(self, vel, rho, dt, rk_order):
+        return self._advect_semi_lagrange_single_channel(
+            vel=vel, rho=rho, dt=dt, rk_order=rk_order, sample_pos="node"
+        )
+
+    def _advect_mac_semi_lagrange_inaccurate(self, vel, rho, dt, rk_order):
+        rho_adv_centered = self._advect_semi_lagrange_single_channel(
+            vel=vel,
+            rho=self.mac_op.get_centered(rho),
+            dt=dt,
+            rk_order=rk_order,
+            sample_pos="center",
+        )
+        rho_adv = self.mac_op.get_mac(rho_adv_centered)
+        return rho_adv
+
+    def _advect_mac_semi_lagrange(self, vel, rho, dt, rk_order):
+        # advect in x direction
+        rho_adv_x = self._advect_semi_lagrange_single_channel(
+            vel=vel, rho=rho[0:1, ...], dt=dt, rk_order=rk_order, sample_pos="mac_x"
+        )
+        # advect in y direction
+        rho_adv_y = self._advect_semi_lagrange_single_channel(
+            vel=vel, rho=rho[1:2, ...], dt=dt, rk_order=rk_order, sample_pos="mac_y"
+        )
+        # concatenate back advected values into the same tensor
+        rho_adv = torch.cat((rho_adv_x, rho_adv_y), dim=0)
+        # advect in z direction in 3d case
+        if self.solver.is_3d():
+            rho_adv_z = self._advect_semi_lagrange_single_channel(
+                vel=vel, rho=rho[2:3, ...], dt=dt, rk_order=rk_order, sample_pos="mac_z"
+            )
+            rho_adv = torch.cat((rho_adv, rho_adv_z), dim=0)
+        return rho_adv
+
+    def _advect_semi_lagrange_single_channel(self, vel, rho, dt, rk_order, sample_pos):
+        # create mesh grid
+        mgrid = utils.create_mesh_grid(
+            rho.size(), device=self.device, dtype=self.solver.dtype
+        )
+        # use runge kutta or euler method to create the back trace field
+        back_trace = self._runge_kutta(vel, mgrid, dt, rk_order, sample_pos)
+        # use warped grid sample function from PyTorch to interpolate the grid
+        # and bring the interporlated values to the current position
+        rho_adv = utils.grid_sample(rho, back_trace)
+        return rho_adv
+
+    def _euler(self, vel, mgrid, dt, sample_pos):
+        if sample_pos == "node":
+            raise NotImplementedError(
+                "Euler step not implemented " "for node data advection."
+            )
+        get_pos_fcn = self._map_sample_pos(sample_pos, rk=False)
+        return mgrid - get_pos_fcn(vel) * dt
+
+    def _runge_kutta(self, vel, mgrid, dt, rk_order, sample_pos):
+        grid_sample_fcn = self._map_sample_pos(sample_pos, rk=True)
+        if rk_order == 1:
+            # return mgrid - grid_sample_fcn(vel, mgrid)*dt
+            return self._euler(vel, mgrid, dt, sample_pos)
+        elif rk_order == 2:
+            # find the mid point position using half of time step
+            # back_trace_mid = mgrid - grid_sample_fcn(vel, mgrid)*dt*0.5
+            back_trace_mid = self._euler(vel, mgrid, 0.5 * dt, sample_pos)
+            # find the mid point velocity
+            return mgrid - grid_sample_fcn(vel, back_trace_mid) * dt
+        elif rk_order == 3:
+            # k1 = grid_sample_fcn(vel, mgrid)
+            k1 = (mgrid - self._euler(vel, mgrid, dt, sample_pos)) / dt
+            k2 = grid_sample_fcn(vel, mgrid - 0.5 * dt * k1)