Feature: Support machine-learning based kinetic energy density functional for OFDFT (#5777)

* refactor: Remove the global variable 'global_pseudo_type' and the input variable 'pseudo_type'.
Now the type of pseudopotential is assigned in STRU, not INPUT, and different type of pseudopotentials can be assigned for different elements in one calculation.
Following are the supported format of STRU:
Al 26.98 Al_ONCV_PBE-1.2.upf
Al 26.98 Al_ONCV_PBE-1.2.upf auto
Al 26.98 Al_ONCV_PBE-1.2.upf upf201
Al 26.98 Al_ONCV_PBE-1.2.upf upf201 # annotations...
If no pseudo_type is assigned in STRU, the default value is 'auto', and ABACUS will recongnize the pp by itself.

* test: Remove `pseudo_type` from INPUT of all integrate tests, and add them into STRU.
Add two integrate tests, `101_PW_upf201_blps_pseudopots` and `101_PW_upf201_upf100_pseudopots`, which contain two elements (Mg3Al), and different types of pseudopotentials are specified for different elements.

* <Feature> Add input parameters about ML_KE

* <Feature> Add files about ML_KE into module_hamilt_pw/hamilt_ofdft

* <Feature> Link MPN KEDF to ABACUS

* <Test> Add a unittest of MPN KEDF: 902_OF_KE_MPN

* <Feature> Output more detailed information of Energy in KSDFT

* <Test> Update the integrate test 902_OF_KE_MPN

* <Feature> In order to check the validity of MPN KEDF, calculate and output TF KEDF when running MPN KEDF.

* Check the input rho in the localTest of MPN KEDF

* <Feature> Add a new descriptor: r_min = min|r-R| / r0, r0 = (Omega / natom) ** (1/3).

* <Fix> Fix the scaling formula of kernel.

* Correct the defination of img.

* Update parameters

* Feature: Add a compile option ENABLE_MLKEDF to control the ML KEDF module.

* Feature: Simplify the code.

* Support Makefile

* Refactor:
Remove GlobalC::ucell.
Optimize the format of input parameters of ML KEDF.

* Refactor:
1. Update annotation.
2. Remove useless parameters and codes.

* Doc: Update documents.

* Test: Add two integrate tests: 902_OF_KE_MPN and 902_OF_KE_CPN5

* Doc: Update annotations.

* Refactor: Update the CMakeLists.txt of nnof

* Test: Configure compilation test.

* [pre-commit.ci lite] apply automatic fixes

* Replace pow with std::pow

* 1. Rename potential.h (class Potential) as pauli_potential.h (class PauliPotential).
2. Replace si.gga.psp used by 902_OF_KE_CPN5 with si.lda.lps.

* Feature: Provide default parameters for MPN and CPN5 KEDF.

* Doc: Update the document of `of_kinetic`.

* Update the file names of nnof

* Make: Update Makefile, and compile documents.

* Remove useless input parameters: `of_ml_nnode` and `of_ml_nlayer`

* Update the integrate tests of MPN and CPN5 KEDFs

* Add function `get_local_pp_energy` into ElecState to calculatet the energy contributed by local pseudopotential.

* Correct the name of eion_elec to elocal_pp, since the nonlocal part is not considered.
Delet the output of Ekinetic, since it only works for local pp.

* Test: Fix module_elecstate/test/elecstate_energy_test.cpp

* [pre-commit.ci lite] apply automatic fixes

* Remove useless head file parallel_reduce.h from esolver_ks.cpp

* Format ml_tools/data.cpp and ml_tools/grid.cpp

* [pre-commit.ci lite] apply automatic fixes

---------

Co-authored-by: pre-commit-ci-lite[bot] <117423508+pre-commit-ci-lite[bot]@users.noreply.github.com>

Author: sunliang98

.github/workflows/build_test_cmake.yml

@@ -17,10 +17,10 @@ jobs:
             name: "Build with Intel toolchain"
 
           - tag: gnu
-            build_args: "-DENABLE_LIBXC=1 -DENABLE_DEEPKS=1 -DENABLE_LIBRI=1 -DENABLE_PAW=1"
+            build_args: "-DENABLE_LIBXC=1 -DENABLE_DEEPKS=1 -DENABLE_MLKEDF=1 -DENABLE_LIBRI=1 -DENABLE_PAW=1"
             name: "Build extra components with GNU toolchain"
           - tag: intel
-            build_args: "-DENABLE_LIBXC=1 -DENABLE_DEEPKS=1 -DENABLE_LIBRI=1"
+            build_args: "-DENABLE_LIBXC=1 -DENABLE_DEEPKS=1 -DENABLE_MLKEDF=1 -DENABLE_LIBRI=1"
             name: "Build extra components with Intel toolchain"
 
           - tag: cuda

.github/workflows/test.yml

@@ -31,7 +31,7 @@ jobs:
 
       - name: Configure
         run: |
-          cmake -B build -DBUILD_TESTING=ON -DENABLE_DEEPKS=ON -DENABLE_LIBXC=ON -DENABLE_LIBRI=ON -DENABLE_PAW=ON -DENABLE_GOOGLEBENCH=ON -DENABLE_RAPIDJSON=ON  -DCMAKE_EXPORT_COMPILE_COMMANDS=1
+          cmake -B build -DBUILD_TESTING=ON -DENABLE_DEEPKS=ON -DENABLE_MLKEDF=ON -DENABLE_LIBXC=ON -DENABLE_LIBRI=ON -DENABLE_PAW=ON -DENABLE_GOOGLEBENCH=ON -DENABLE_RAPIDJSON=ON  -DCMAKE_EXPORT_COMPILE_COMMANDS=1
 
       - uses: pre-commit/[email protected]
         with:

CMakeLists.txt

@@ -12,6 +12,7 @@ project(
 
 option(ENABLE_LCAO "Enable LCAO calculation." ON)
 option(ENABLE_DEEPKS "Enable DeePKS functionality" OFF)
+option(ENABLE_MLKEDF "Enable Machine Learning based KEDF for OFDFT" OFF)
 option(ENABLE_LIBXC "Enable LibXC functionality" OFF)
 option(USE_CUDA "Enable support to CUDA for ABACUS." OFF)
 option(ENABLE_FLOAT_FFTW "Enable support to single precision FFTW library." OFF)
@@ -469,10 +470,29 @@ if(ENABLE_DEEPKS)
   add_compile_definitions(__DEEPKS)
 endif()
 
+if(ENABLE_MLKEDF)
+  target_link_libraries(${ABACUS_BIN_NAME} hamilt_mlkedf)
+
+  find_path(libnpy_SOURCE_DIR npy.hpp HINTS ${libnpy_INCLUDE_DIR})
+  if(NOT libnpy_SOURCE_DIR)
+    include(FetchContent)
+    FetchContent_Declare(
+      libnpy
+      GIT_REPOSITORY https://github.com/llohse/libnpy.git
+      GIT_SHALLOW TRUE
+      GIT_PROGRESS TRUE)
+    FetchContent_MakeAvailable(libnpy)
+  else()
+    include_directories(${libnpy_INCLUDE_DIR})
+  endif()
+  include_directories(${libnpy_SOURCE_DIR}/include)
+  add_compile_definitions(__MLKEDF)
+endif()
+
 # Torch uses outdated components to detect CUDA arch, causing failure on
 # latest CUDA kits. Set CMake variable TORCH_CUDA_ARCH_LIST in the form of
 # "major.minor" if required.
-if(ENABLE_DEEPKS OR DEFINED Torch_DIR)
+if(ENABLE_DEEPKS OR ENABLE_MLKEDF OR DEFINED Torch_DIR)
   find_package(Torch REQUIRED)
   if(NOT Torch_VERSION VERSION_LESS "2.1.0")
     set_if_higher(CMAKE_CXX_STANDARD 17)

docs/CITATIONS.md

@@ -33,3 +33,9 @@ The following references are required to be cited when using ABACUS. Specificall
     Peize Lin, Xinguo Ren, and Lixin He. "Efficient Hybrid Density Functional Calculations for Large Periodic Systems Using Numerical Atomic Orbitals." Journal of Chemical Theory and Computation 2021, 17(1), 222–239.
 
     Peize Lin, Xinguo Ren, and Lixin He. "Accuracy of Localized Resolution of the Identity in Periodic Hybrid Functional Calculations with Numerical Atomic Orbitals." Journal of Physical Chemistry Letters 2020, 11, 3082-3088.
+
+- **If ML-KEDF is used:**
+
+    Sun, Liang, and Mohan Chen. "Machine learning based nonlocal kinetic energy density functional for simple metals and alloys." Physical Review B 109.11 (2024): 115135.
+
+    Sun, Liang, and Mohan Chen. "Multi-channel machine learning based nonlocal kinetic energy density functional for semiconductors." Electronic Structure 6.4 (2024): 045006.

docs/advanced/input_files/input-main.md

@@ -210,6 +210,36 @@
     - [of\_kernel\_file](#of_kernel_file)
     - [of\_full\_pw](#of_full_pw)
     - [of\_full\_pw\_dim](#of_full_pw_dim)
+  - [ML-KEDF: machine learning based kinetic energy density functional for OFDFT](#ml-kedf-machine-learning-based-kinetic-energy-density-functional-for-ofdft)
+    - [of\_ml\_gene\_data](#of_ml_gene_data)
+    - [of\_ml\_device](#of_ml_device)
+    - [of\_ml\_feg](#of_ml_feg)
+    - [of\_ml\_nkernel](#of_ml_nkernel)
+    - [of\_ml\_kernel](#of_ml_kernel)
+    - [of\_ml\_kernel\_scaling](#of_ml_kernel_scaling)
+    - [of\_ml\_yukawa\_alpha](#of_ml_yukawa_alpha)
+    - [of\_ml\_kernel\_file](#of_ml_kernel_file)
+    - [of\_ml\_gamma](#of_ml_gamma)
+    - [of\_ml\_p](#of_ml_p)
+    - [of\_ml\_q](#of_ml_q)
+    - [of\_ml\_tanhp](#of_ml_tanhp)
+    - [of\_ml\_tanhq](#of_ml_tanhq)
+    - [of\_ml\_gammanl](#of_ml_gammanl)
+    - [of\_ml\_pnl](#of_ml_pnl)
+    - [of\_ml\_qnl](#of_ml_qnl)
+    - [of\_ml\_xi](#of_ml_xi)
+    - [of\_ml\_tanhxi](#of_ml_tanhxi)
+    - [of\_ml\_tanhxi\_nl](#of_ml_tanhxi_nl)
+    - [of\_ml\_tanh\_pnl](#of_ml_tanh_pnl)
+    - [of\_ml\_tanh\_qnl](#of_ml_tanh_qnl)
+    - [of\_ml\_tanhp\_nl](#of_ml_tanhp_nl)
+    - [of\_ml\_tanhq\_nl](#of_ml_tanhq_nl)
+    - [of\_ml\_chi\_p](#of_ml_chi_p)
+    - [of\_ml\_chi\_q](#of_ml_chi_q)
+    - [of\_ml\_chi\_xi](#of_ml_chi_xi)
+    - [of\_ml\_chi\_pnl](#of_ml_chi_pnl)
+    - [of\_ml\_chi\_qnl](#of_ml_chi_qnl)
+    - [of\_ml\_local\_test](#of_ml_local_test)
   - [Electric field and dipole correction](#electric-field-and-dipole-correction)
     - [efield\_flag](#efield_flag)
     - [dip\_cor\_flag](#dip_cor_flag)
@@ -2093,11 +2123,18 @@ Warning: this function is not robust enough for the current version. Please try
 - **Type**: String
 - **Availability**: OFDFT
 - **Description**: The type of KEDF (kinetic energy density functional).
+
+  Analytical functionals:
   - **wt**: Wang-Teter.
   - **tf**: Thomas-Fermi.
   - **vw**: von Weizsäcker.
-  - **tf+**: TF$\rm{\lambda}$vW, the parameter $\rm{\lambda}$ can be set by `of_vw_weight`.
+  - **tf+**: TF $\rm{\lambda}$ vW, the parameter $\rm{\lambda}$ can be set by `of_vw_weight`.
   - **lkt**: Luo-Karasiev-Trickey.
+
+  Machine learning (ML) based functionals:
+  - **ml**: ML-based KEDF allows for greater flexibility, enabling users to set related ML model parameters themselves. see [ML-KEDF: machine learning based kinetic energy density functional for OFDFT](#ml-kedf-machine-learning-based-kinetic-energy-density-functional-for-ofdft).
+  - **mpn**: ML-based Physically-constrained Non-local (MPN) KEDF. ABACUS automatically configures the necessary parameters, requiring no manual intervention from the user.
+  - **cpn5**: Multi-Channel MPN (CPN) KEDF with 5 channels. Similar to mpn, ABACUS handles all parameter settings automatically.
 - **Default**: wt
 
 ### of_method
@@ -2228,6 +2265,221 @@ Warning: this function is not robust enough for the current version. Please try
 
 [back to top](#full-list-of-input-keywords)
 
+## ML-KEDF: machine learning based kinetic energy density functional for OFDFT
+
+### of_ml_gene_data
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Generate training data or not.
+- **Default**: False
+
+### of_ml_device
+
+- **Type**: String
+- **Availability**: OFDFT
+- **Description**: Run Neural Network on GPU or CPU.
+  - **cpu**: CPU
+  - **gpu**: GPU
+- **Default**: cpu
+
+### of_ml_feg
+
+- **Type**: Integer
+- **Availability**: OFDFT
+- **Description**: The method to incorporate the Free Electron Gas (FEG) limit: $F_\theta |_{\rm{FEG}} = 1$, where $F_\theta$ is enhancement factor of Pauli energy.
+  - **0**: Do not incorporate the FEG limit.
+  - **1**: Incorporate the FEG limit by translation: $F_\theta = F^{\rm{NN}}_\theta - F^{\rm{NN}}_\theta|_{\rm{FEG}} + 1$.
+  - **3**: Incorporate the FEG limit by nonlinear transformation: $F_\theta = f(F^{\rm{NN}}_\theta - F^{\rm{NN}}_\theta|_{\rm{FEG}} + \ln(e - 1))$, where $f = \ln(1 + e^x)$ is softplus function, satisfying $f(x)|_{x=\ln(e-1)} = 1$. 
+- **Default**: 0
+
+### of_ml_nkernel
+
+- **Type**: Integer
+- **Availability**: OFDFT
+- **Description**: Number of kernel functions.
+- **Default**: 1
+
+### of_ml_kernel
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the type of the $i$-th kernel function.
+  - **1**: Wang-Teter kernel function.
+  - **2**: Modified Yukawa function: $k_{\rm{F}}^2\frac{\exp{({-\alpha k_{\rm{F}}|\mathbf{r}-\mathbf{r}'|})}}{|\mathbf{r}-\mathbf{r}'|}$, and $\alpha$ is specified by [of_ml_yukawa_alpha](#of_ml_yukawa_alpha).
+  - **3**: Truncated kinetic kernel (TKK), the file containing TKK is specified by [of_ml_kernel_file](#of_kernel_file).
+- **Default**: 1
+
+### of_ml_kernel_scaling
+
+- **Type**: Vector of Real
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the RECIPROCAL of scaling parameter $\lambda$ of the $i$-th kernel function. $w_i(\mathbf{r}-\mathbf{r}') = \lambda^3 w_i'(\lambda(\mathbf{r}-\mathbf{r}'))$.
+- **Default**: 1.0
+
+### of_ml_yukawa_alpha
+
+- **Type**: Vector of Real
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the parameter $\alpha$ of $i$-th kernel function. ONLY used for Yukawa kernel function.
+- **Default**: 1.0
+
+### of_ml_kernel_file
+
+- **Type**: Vector of String
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the file containint the $i$-th kernel function. ONLY used for TKK.
+- **Default**: none
+
+### of_ml_gamma
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Local descriptor: $\gamma(\mathbf{r}) = (\rho(\mathbf{r}) / \rho_0)^{1/3}$.
+- **Default**: False
+
+### of_ml_p
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Semi-local descriptor: $p(\mathbf{r}) = \frac{|\nabla \rho(\mathbf{r})|^2} {[2 (3 \pi^2)^{1/3} \rho^{4/3}(\mathbf{r})]^2}$.
+- **Default**: False
+
+### of_ml_q
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Semi-local descriptor: $q(\mathbf{r}) = \frac{\nabla^2 \rho(\mathbf{r})} {[4 (3 \pi^2)^{2/3} \rho^{5/3}(\mathbf{r})]}$.
+- **Default**: False
+
+### of_ml_tanhp
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Semi-local descriptor: $\tilde{p}(\mathbf{r}) = \tanh(\chi_p p(\mathbf{r}))$.
+- **Default**: False
+
+### of_ml_tanhq
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: Semi-local descriptor: $\tilde{q}(\mathbf{r}) = \tanh(\chi_q q(\mathbf{r}))$.
+- **Default**: False
+
+### of_ml_chi_p
+
+- **Type**: Real
+- **Availability**: OFDFT
+- **Description**: Hyperparameter $\chi_p$: $\tilde{p}(\mathbf{r}) = \tanh(\chi_p p(\mathbf{r}))$.
+- **Default**: 1.0
+
+### of_ml_chi_q
+
+- **Type**: Real
+- **Availability**: OFDFT
+- **Description**: Hyperparameter $\chi_q$: $\tilde{q}(\mathbf{r}) = \tanh(\chi_q q(\mathbf{r}))$.
+- **Default**: False
+
+### of_ml_gammanl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\gamma_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') \gamma(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_pnl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $p_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') p(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_qnl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $q_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') q(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_xi
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\xi(\mathbf{r}) = \frac{\int{w_i(\mathbf{r}-\mathbf{r}') \rho^{1/3}(\mathbf{r}') dr'}}{\rho^{1/3}(\mathbf{r})}$. 
+- **Default**: 0
+
+### of_ml_tanhxi
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{\xi}(\mathbf{r}) = \tanh(\chi_{\xi} \xi(\mathbf{r}))$. 
+- **Default**: 0
+
+### of_ml_tanhxi_nl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{\xi}_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') \tilde{\xi}(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_tanh_pnl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{p_{\rm{nl}}}(\mathbf{r}) = \tanh{(\chi_{p_{\rm{nl}}} p_{\rm{nl}}(\mathbf{r}))}$. 
+- **Default**: 0
+
+### of_ml_tanh_qnl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{q_{\rm{nl}}}(\mathbf{r}) = \tanh{(\chi_{q_{\rm{nl}}} q_{\rm{nl}}(\mathbf{r}))}$. 
+- **Default**: 0
+
+### of_ml_tanhp_nl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{p}_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') \tilde{p}(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_tanhq_nl
+
+- **Type**: Vector of Integer 
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element controls the non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{q}_{\rm{nl}}(\mathbf{r}) = \int{w_i(\mathbf{r}-\mathbf{r}') \tilde{q}(\mathbf{r}') dr'}$. 
+- **Default**: 0
+
+### of_ml_chi_xi
+
+- **Type**: Vector of Real
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the hyperparameter $\chi_\xi$ of non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{\xi}(\mathbf{r}) = \tanh(\chi_{\xi} \xi(\mathbf{r}))$. 
+- **Default**: 1.0
+
+### of_ml_chi_pnl
+
+- **Type**: Vector of Real
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the hyperparameter $\chi_{p_{\rm{nl}}}$ of non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{p_{\rm{nl}}}(\mathbf{r}) = \tanh{(\chi_{p_{\rm{nl}}} p_{\rm{nl}}(\mathbf{r}))}$. 
+- **Default**: 1.0
+
+### of_ml_chi_qnl
+
+- **Type**: Vector of Real
+- **Availability**: OFDFT
+- **Description**: Containing nkernel (see [of_ml_nkernel](#of_ml_nkernel)) elements. The $i$-th element specifies the hyperparameter $\chi_{q_{\rm{nl}}}$ of non-local descriptor defined by the $i$-th kernel function $w_i(\mathbf{r}-\mathbf{r}')$: $\tilde{q_{\rm{nl}}}(\mathbf{r}) = \tanh{(\chi_{q_{\rm{nl}}} q_{\rm{nl}}(\mathbf{r}))}$. 
+- **Default**: 1.0
+
+### of_ml_local_test
+
+- **Type**: Boolean
+- **Availability**: OFDFT
+- **Description**: FOR TEST. Read in the density, and output the F and Pauli potential.
+- **Default**: False
+
+[back to top](#full-list-of-input-keywords)
+
 ## Electric field and dipole correction
 
 These variables are relevant to electric field and dipole correction