diff --git a/paddle/fluid/operators/p_norm_op.cc b/paddle/fluid/operators/p_norm_op.cc
new file mode 100644
index 0000000000000000000000000000000000000000..1da6b11935b3aa09884b40ebfb3448581d9b34f9
--- /dev/null
+++ b/paddle/fluid/operators/p_norm_op.cc
@@ -0,0 +1,139 @@
+/* Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+Indicesou may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License. */
+
+#include "paddle/fluid/operators/p_norm_op.h"
+#include <memory>
+#include <string>
+#include <vector>
+
+namespace paddle {
+namespace operators {
+
+class PnormOpMaker : public framework::OpProtoAndCheckerMaker {
+ public:
+  void Make() override {
+    AddInput("X", "(Tensor) A tensor of rank >= axis.");
+    AddAttr<float>("porder",
+                   "The porder is the p order vector norm to calculate.")
+        .SetDefault(2.0f);
+    AddAttr<int>("axis",
+                 "The axis on which to apply normalization. If axis < 0, "
+                 "the dimension to pnorm is rank(X) + axis. -1 is "
+                 "the last dimension.")
+        .SetDefault(-1);
+    AddAttr<float>("epsilon",
+                   "(float, default 1e-10) The epsilon value is used "
+                   "to avoid division by zero.")
+        .SetDefault(1.0e-12f);
+    AddAttr<bool>(
+        "keepdim",
+        "(bool, default false) Whether to keep the dimensions as the input")
+        .SetDefault(false);
+    AddOutput(
+        "Out",
+        "(Tensor) Output tensor for the `(sum(x.pow(p)) + epsion).pow(1/p)`");
+    AddComment(R"DOC(
+
+Given a tensor, apply 2-normalization along the provided axis.
+
+$$
+pnorm = \(\sum_i {abs\(x_i\)^p}  \)^{1/p}
+$$
+
+where, $\sum_i{x_i^p}$ is calculated along the `axis` dimension.
+        
+)DOC");
+  }
+};
+
+class PnormOp : public framework::OperatorWithKernel {
+ public:
+  using framework::OperatorWithKernel::OperatorWithKernel;
+  void InferShape(framework::InferShapeContext* ctx) const override {
+    OP_INOUT_CHECK(ctx->HasInput("X"), "Input", "X", "p_norm");
+    OP_INOUT_CHECK(ctx->HasOutput("Out"), "Output", "Out", "p_norm");
+    auto porder = ctx->Attrs().Get<float>("porder");
+    PADDLE_ENFORCE_NE(porder, 0,
+                      platform::errors::InvalidArgument(
+                          "The input porder of p_norm is not support for "
+                          "porder == 0, INFINITY, -INFINITY now."));
+    PADDLE_ENFORCE_NE(porder, INFINITY,
+                      platform::errors::InvalidArgument(
+                          "The input porder of p_norm is not support for "
+                          "porder == 0, INFINITY, -INFINITY now."));
+    PADDLE_ENFORCE_NE(porder, -INFINITY,
+                      platform::errors::InvalidArgument(
+                          "The input porder of p_norm is not support for "
+                          "porder == 0, INFINITY, -INFINITY now."));
+    auto xdim = ctx->GetInputDim("X");
+    int axis = ctx->Attrs().Get<int>("axis");
+    bool keepdim = ctx->Attrs().Get<bool>("keepdim");
+    if (axis < 0) axis = xdim.size() + axis;
+    std::vector<int> reduce_dims;
+    for (int i = 0; i < xdim.size(); ++i) {
+      if (i != axis) reduce_dims.emplace_back(xdim[i]);
+    }
+    xdim[axis] = 1;
+    if (keepdim) {
+      ctx->SetOutputDim("Out", xdim);
+    } else {
+      ctx->SetOutputDim("Out", framework::make_ddim(reduce_dims));
+    }
+  }
+};
+
+class PnormOpGrad : public framework::OperatorWithKernel {
+ public:
+  using framework::OperatorWithKernel::OperatorWithKernel;
+  void InferShape(framework::InferShapeContext* ctx) const override {
+    OP_INOUT_CHECK(ctx->HasInput("X"), "Input", "X", "p_norm");
+    OP_INOUT_CHECK(ctx->HasInput("Out"), "Input", "Out", "p_norm");
+    OP_INOUT_CHECK(ctx->HasInput(framework::GradVarName("Out")), "Input",
+                   "Out@GRAD", "p_norm");
+    OP_INOUT_CHECK(ctx->HasOutput(framework::GradVarName("X")), "Output",
+                   "X@GRAD", "p_norm");
+    ctx->SetOutputDim(framework::GradVarName("X"), ctx->GetInputDim("X"));
+  }
+};
+
+template <typename T>
+class PnormOpGradOpMaker : public framework::SingleGradOpMaker<T> {
+ public:
+  using framework::SingleGradOpMaker<T>::SingleGradOpMaker;
+
+ protected:
+  void Apply(GradOpPtr<T> op) const override {
+    op->SetType("p_norm_grad");
+    op->SetAttrMap(this->Attrs());
+    op->SetInput("X", this->Input("X"));
+    op->SetInput("Out", this->Output("Out"));
+    op->SetInput(framework::GradVarName("Out"), this->OutputGrad("Out"));
+    op->SetOutput(framework::GradVarName("X"), this->InputGrad("X"));
+  }
+};
+
+}  // namespace operators
+}  // namespace paddle
+
+namespace ops = paddle::operators;
+using CPU = paddle::platform::CPUDeviceContext;
+
+REGISTER_OPERATOR(p_norm, ops::PnormOp, ops::PnormOpMaker,
+                  ops::PnormOpGradOpMaker<paddle::framework::OpDesc>,
+                  ops::PnormOpGradOpMaker<paddle::imperative::OpBase>);
+REGISTER_OPERATOR(p_norm_grad, ops::PnormOpGrad);
+REGISTER_OP_CPU_KERNEL(p_norm, ops::PnormKernel<CPU, float>,
+                       ops::PnormKernel<CPU, double>);
+REGISTER_OP_CPU_KERNEL(p_norm_grad, ops::PnormGradKernel<CPU, float>,
+                       ops::PnormGradKernel<CPU, double>);
diff --git a/paddle/fluid/operators/p_norm_op.cu b/paddle/fluid/operators/p_norm_op.cu
new file mode 100644
index 0000000000000000000000000000000000000000..5e94d87b85197b69fb47643168facb8de6375884
--- /dev/null
+++ b/paddle/fluid/operators/p_norm_op.cu
@@ -0,0 +1,180 @@
+/* Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+Indicesou may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License. */
+
+#include <algorithm>
+#include "cub/cub.cuh"
+#include "paddle/fluid/operators/p_norm_op.h"
+
+namespace paddle {
+namespace operators {
+
+template <typename T>
+__device__ __forceinline__ int sgn(T val) {
+  return (T(0) < val) - (val < T(0));
+}
+
+__device__ __forceinline__ float inline_abs(float x) { return abs(x); }
+__device__ __forceinline__ double inline_abs(double x) { return abs(x); }
+
+__device__ __forceinline__ int inline_sign(float x) { return sgn<float>(x); }
+__device__ __forceinline__ int inline_sign(double x) { return sgn<double>(x); }
+
+__device__ __forceinline__ float inline_pow(float base, float exponent) {
+  return pow(base, exponent);
+}
+__device__ __forceinline__ double inline_pow(double base, double exponent) {
+  return pow(base, exponent);
+}
+
+template <typename T, int BlockDim>
+__global__ void Pnorm(const T* x, const int pre,
+                      const int axis_n,  // dim in axis
+                      const int post, float porder, T* out_norm) {
+  typedef cub::BlockReduce<T, BlockDim> BlockReduce;
+  __shared__ typename BlockReduce::TempStorage temp_storage;
+  int num = pre * post;
+  for (int i = blockIdx.x; i < num; i += gridDim.x) {
+    int base = (i / post) * post * axis_n + (i % post);
+
+    T sum = 0.0;
+    __shared__ T norm;
+    for (int j = threadIdx.x; j < axis_n; j += blockDim.x) {
+      const T x_ij = x[base + j * post];
+      sum += inline_pow(inline_abs(x_ij), porder);
+    }
+    T reduce_result = BlockReduce(temp_storage).Sum(sum);
+
+    if (threadIdx.x == 0) {
+      norm = inline_pow(reduce_result, 1.0f / porder);
+      out_norm[i] = norm;
+    }
+    __syncthreads();
+  }
+}
+
+template <typename DeviceContext, typename T>
+class PnormCUDAKernel : public framework::OpKernel<T> {
+ public:
+  void Compute(const framework::ExecutionContext& ctx) const override {
+    auto* in_x = ctx.Input<framework::Tensor>("X");
+    auto* out_norm = ctx.Output<framework::Tensor>("Out");
+    const T* x = in_x->data<T>();
+    T* norm = out_norm->mutable_data<T>(ctx.GetPlace());
+
+    auto xdim = in_x->dims();
+    auto ndim = out_norm->dims();
+    float porder = ctx.Attr<float>("porder");
+    int axis = ctx.Attr<int>("axis");
+    if (axis < 0) axis = xdim.size() + axis;
+    int pre, n, post;
+    GetDims(xdim, axis, &pre, &n, &post);
+
+    auto& dev_ctx = ctx.cuda_device_context();
+
+    const int block = 512;
+    int max_threads = dev_ctx.GetMaxPhysicalThreadCount();
+    const int max_blocks = std::max(max_threads / block, 1);
+    int grid = std::min(max_blocks, pre * post);
+    Pnorm<T, block><<<grid, block, 0, dev_ctx.stream()>>>(x, pre, n, post,
+                                                          porder, norm);
+  }
+};
+
+template <typename T, int BlockDim>
+__global__ void PnormGradient(const T* x, const T* x_norm, const T* y_grad,
+                              const float porder, const int pre,
+                              const int axis_n, const int post, const T eps,
+                              T* x_grad) {
+  typedef cub::BlockReduce<T, BlockDim> BlockReduce;
+  __shared__ typename BlockReduce::TempStorage temp_storage_sum;
+  // dx = (x/pnorm_broadcast).pow(p-1) * norm_dy.broadcast * sign(x)
+  int num = pre * post;
+  for (int i = blockIdx.x; i < num; i += gridDim.x) {
+    T sum = 0.0;
+    __shared__ T row_sum;
+    __shared__ T row_sqrt_norm;
+    __shared__ T row_norm;
+
+    auto base = (i / post) * post * axis_n + (i % post);
+
+    for (int j = threadIdx.x; j < axis_n; j += blockDim.x) {
+      int index = base + j * post;
+      sum += x[index] * y_grad[index];
+    }
+    T reduce_result = BlockReduce(temp_storage_sum).Sum(sum);
+
+    if (threadIdx.x == 0) {
+      row_sum = reduce_result;
+      row_sqrt_norm = x_norm[i];
+      row_norm = row_sqrt_norm * row_sqrt_norm;
+    }
+    __syncthreads();
+
+    const T pnorm_i = x_norm[i];
+    const T yout_i = y_grad[i];
+
+    for (int j = threadIdx.x; j < axis_n; j += blockDim.x) {
+      int index = base + j * post;
+      const T x_ij = inline_abs(x[index]);
+      const T dy_ij = y_grad[index];
+      x_grad[index] = inline_pow(x_ij, porder - 1.0f) /
+                      (inline_pow(pnorm_i, porder - 1.0f) + eps) * yout_i *
+                      inline_sign(x[index]);
+    }
+  }
+}
+
+template <typename DeviceContext, typename T, typename AttrType = T>
+class PnormGradCUDAKernel : public framework::OpKernel<T> {
+ public:
+  void Compute(const framework::ExecutionContext& ctx) const override {
+    auto* in_x = ctx.Input<framework::Tensor>("X");
+    auto* in_norm = ctx.Input<framework::Tensor>("Out");
+    auto* in_norm_dy =
+        ctx.Input<framework::Tensor>(framework::GradVarName("Out"));
+    auto* out_dx = ctx.Output<framework::Tensor>(framework::GradVarName("X"));
+    T* dx = out_dx->mutable_data<T>(ctx.GetPlace());
+    const T* x = in_x->data<T>();
+    const T* x_norm = in_norm->data<T>();
+    const T* norm_dy = in_norm_dy->data<T>();
+
+    auto xdim = in_x->dims();
+    float porder = ctx.Attr<float>("porder");
+    T eps = static_cast<T>(ctx.Attr<float>("epsilon"));
+    int axis = ctx.Attr<int>("axis");
+    if (axis < 0) axis = xdim.size() + axis;
+    int pre, n, post;
+    GetDims(xdim, axis, &pre, &n, &post);
+
+    auto& dev_ctx = ctx.cuda_device_context();
+
+    const int block = 512;
+    int max_threads = dev_ctx.GetMaxPhysicalThreadCount();
+    const int max_blocks = std::max(max_threads / block, 1);
+    int grid = std::min(max_blocks, pre * post);
+    PnormGradient<T, block><<<grid, block, 0, dev_ctx.stream()>>>(
+        x, x_norm, norm_dy, porder, pre, n, post, eps, dx);
+  }
+};
+
+}  // namespace operators
+}  // namespace paddle
+
+namespace ops = paddle::operators;
+using CUDA = paddle::platform::CUDADeviceContext;
+
+REGISTER_OP_CUDA_KERNEL(p_norm, ops::PnormCUDAKernel<CUDA, float>,
+                        ops::PnormCUDAKernel<CUDA, double>);
+REGISTER_OP_CUDA_KERNEL(p_norm_grad, ops::PnormGradCUDAKernel<CUDA, float>,
+                        ops::PnormGradCUDAKernel<CUDA, double>);
diff --git a/paddle/fluid/operators/p_norm_op.h b/paddle/fluid/operators/p_norm_op.h
new file mode 100644
index 0000000000000000000000000000000000000000..c5bdfe352723b55f80376d6644922af5de099e90
--- /dev/null
+++ b/paddle/fluid/operators/p_norm_op.h
@@ -0,0 +1,112 @@
+/* Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+Indicesou may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License. */
+
+#pragma once
+#include "paddle/fluid/framework/op_registry.h"
+#include "paddle/fluid/operators/math/math_function.h"
+
+namespace paddle {
+namespace operators {
+
+inline void GetDims(const framework::DDim& dim, int axis, int* pre, int* n,
+                    int* post) {
+  *pre = 1;
+  *post = 1;
+  *n = dim[axis];
+  for (int i = 0; i < axis; ++i) {
+    (*pre) *= dim[i];
+  }
+  for (int i = axis + 1; i < dim.size(); ++i) {
+    (*post) *= dim[i];
+  }
+}
+
+template <typename DeviceContext, typename T>
+class PnormKernel : public framework::OpKernel<T> {
+ public:
+  void Compute(const framework::ExecutionContext& ctx) const override {
+    auto* in_x = ctx.Input<framework::Tensor>("X");
+    auto* out_norm = ctx.Output<framework::Tensor>("Out");
+    out_norm->mutable_data<T>(ctx.GetPlace());
+
+    auto xdim = in_x->dims();
+    float porder = ctx.Attr<float>("porder");
+    int axis = ctx.Attr<int>("axis");
+    if (axis < 0) axis = xdim.size() + axis;
+    int pre, n, post;
+    GetDims(xdim, axis, &pre, &n, &post);
+
+    auto* place = ctx.template device_context<DeviceContext>().eigen_device();
+
+    Eigen::DSizes<int, 3> shape(pre, n, post);
+    Eigen::DSizes<int, 2> norm_shape(pre, post);
+
+    auto x_e = framework::EigenVector<T>::Flatten(*in_x);
+    auto norm_e = framework::EigenVector<T>::Flatten(*out_norm);
+
+    auto x = x_e.reshape(shape);
+    auto norm = norm_e.reshape(norm_shape);
+
+    Eigen::DSizes<int, 1> rdim(1);
+    auto xp = (x.abs()).pow(porder);
+    auto sum = xp.sum(rdim);
+    norm.device(*place) = sum.pow(1.0f / porder);
+  }
+};
+
+template <typename DeviceContext, typename T, typename AttrType = T>
+class PnormGradKernel : public framework::OpKernel<T> {
+ public:
+  void Compute(const framework::ExecutionContext& ctx) const override {
+    auto* in_x = ctx.Input<framework::Tensor>("X");
+    auto* in_norm = ctx.Input<framework::Tensor>("Out");
+    auto* in_norm_dy =
+        ctx.Input<framework::Tensor>(framework::GradVarName("Out"));
+    auto* out_dx = ctx.Output<framework::Tensor>(framework::GradVarName("X"));
+    out_dx->mutable_data<T>(ctx.GetPlace());
+
+    T eps = static_cast<T>(ctx.Attr<float>("epsilon"));
+    auto xdim = in_x->dims();
+    float porder = ctx.Attr<float>("porder");
+
+    int axis = ctx.Attr<int>("axis");
+    if (axis < 0) axis = xdim.size() + axis;
+    int pre, n, post;
+    GetDims(xdim, axis, &pre, &n, &post);
+    Eigen::DSizes<int, 3> shape(pre, n, post);
+    Eigen::DSizes<int, 3> rshape(pre, 1, post);
+
+    auto* place = ctx.template device_context<DeviceContext>().eigen_device();
+
+    auto x_e = framework::EigenVector<T>::Flatten(*in_x);
+    auto dx_e = framework::EigenVector<T>::Flatten(*out_dx);
+    auto norm_e = framework::EigenVector<T>::Flatten(*in_norm);
+    auto norm_dy_e = framework::EigenVector<T>::Flatten(*in_norm_dy);
+
+    auto x = x_e.reshape(shape);
+    auto dx = dx_e.reshape(shape);
+    auto norm = norm_e.reshape(rshape);
+    auto norm_dy = norm_dy_e.reshape(rshape);
+
+    Eigen::DSizes<int, 1> rdim(1);
+    Eigen::DSizes<int, 3> bcast(1, n, 1);
+
+    dx.device(*place) = (x.abs()).pow(porder - 1.0f);
+    dx.device(*place) =
+        dx / ((norm.broadcast(bcast)).pow(porder - 1.0f) + x.constant(eps));
+    dx.device(*place) = dx * norm_dy.broadcast(bcast) * x.sign();
+  }
+};
+}  // namespace operators
+}  // namespace paddle
diff --git a/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cc b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cc
new file mode 100644
index 0000000000000000000000000000000000000000..57df2664824d478503fce04f09c5a7f1e02eb080
--- /dev/null
+++ b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cc
@@ -0,0 +1,65 @@
+// Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#include "paddle/fluid/operators/reduce_ops/frobenius_norm_op.h"
+#include <memory>
+#include <string>
+
+namespace paddle {
+namespace operators {
+
+template <typename T>
+class FrobeniusNormOpGradMaker : public framework::SingleGradOpMaker<T> {
+ public:
+  using framework::SingleGradOpMaker<T>::SingleGradOpMaker;
+
+ protected:
+  void Apply(GradOpPtr<T> op) const override {
+    op->SetType("frobenius_norm_grad");
+    op->SetInput("X", this->Input("X"));
+    op->SetInput("Out", this->Output("Out"));
+    op->SetInput(framework::GradVarName("Out"), this->OutputGrad("Out"));
+    op->SetOutput(framework::GradVarName("X"), this->InputGrad("X"));
+    op->SetAttrMap(this->Attrs());
+  }
+};
+
+}  // namespace operators
+}  // namespace paddle
+
+class FrobeniusNormOpMaker : public ops::ReduceOpMaker {
+ protected:
+  virtual std::string GetName() const { return "frobenius_norm"; }
+  virtual std::string GetOpType() const { return "Reduce frobenius_norm"; }
+};
+
+REGISTER_OPERATOR(frobenius_norm, ops::ReduceOp, FrobeniusNormOpMaker,
+                  ops::FrobeniusNormOpGradMaker<paddle::framework::OpDesc>,
+                  ops::FrobeniusNormOpGradMaker<paddle::imperative::OpBase>);
+
+REGISTER_OPERATOR(frobenius_norm_grad, ops::ReduceGradOp);
+
+REGISTER_OP_CPU_KERNEL(frobenius_norm,
+                       ops::ReduceKernel<paddle::platform::CPUDeviceContext,
+                                         float, ops::FrobeniusNormFunctor>,
+                       ops::ReduceKernel<paddle::platform::CPUDeviceContext,
+                                         double, ops::FrobeniusNormFunctor>);
+
+template <typename T>
+using CPUFrobeniusNormGradKernel =
+    ops::FrobeniusNormGradKernel<paddle::platform::CPUDeviceContext, T,
+                                 ops::FrobeniusNormGradFunctor>;
+
+REGISTER_OP_CPU_KERNEL(frobenius_norm_grad, CPUFrobeniusNormGradKernel<float>,
+                       CPUFrobeniusNormGradKernel<double>);
diff --git a/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cu b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cu
new file mode 100644
index 0000000000000000000000000000000000000000..1ff645dfeb653c5fafa2ae2ca058e780a93a0764
--- /dev/null
+++ b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.cu
@@ -0,0 +1,32 @@
+// Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#include "paddle/fluid/operators/reduce_ops/cub_reduce.h"
+#include "paddle/fluid/operators/reduce_ops/frobenius_norm_op.h"
+
+template <typename T>
+using CUDAFrobeniusNormKernel =
+    ops::ReduceKernel<paddle::platform::CUDADeviceContext, T,
+                      ops::FrobeniusNormFunctor>;
+
+REGISTER_OP_CUDA_KERNEL(frobenius_norm, CUDAFrobeniusNormKernel<float>,
+                        CUDAFrobeniusNormKernel<double>);
+
+template <typename T>
+using CUDAFrobeniusNormGradKernel =
+    ops::ReduceGradKernel<paddle::platform::CUDADeviceContext, T,
+                          ops::FrobeniusNormGradFunctor>;
+
+REGISTER_OP_CUDA_KERNEL(frobenius_norm_grad, CUDAFrobeniusNormGradKernel<float>,
+                        CUDAFrobeniusNormGradKernel<double>);
diff --git a/paddle/fluid/operators/reduce_ops/frobenius_norm_op.h b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.h
new file mode 100644
index 0000000000000000000000000000000000000000..0b6b87d99ecd98e65c492fb96f3a1e886b7bfa4b
--- /dev/null
+++ b/paddle/fluid/operators/reduce_ops/frobenius_norm_op.h
@@ -0,0 +1,54 @@
+// Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//     http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#pragma once
+
+#include <vector>
+
+#include "paddle/fluid/operators/reduce_ops/reduce_op.h"
+
+namespace paddle {
+namespace operators {
+
+// \partial \| X \|_F = \frac{X}{ \| X \|_F }
+template <typename DeviceContext, typename T, typename Functor>
+class FrobeniusNormGradKernel : public framework::OpKernel<T> {
+ public:
+  void Compute(const framework::ExecutionContext& context) const override {
+    // default use Eigen broadcast
+    ReduceGradKernel<DeviceContext, T, Functor, false> kernel;
+    kernel.Compute(context);
+  }
+};
+
+struct FrobeniusNormFunctor {
+  template <typename DeviceContext, typename X, typename Y, typename Dim>
+  void operator()(const DeviceContext& place, X* x, Y* y, const Dim& dim) {
+    y->device(place) = ((x->square()).sum(dim)).sqrt();
+  }
+};
+
+struct FrobeniusNormGradFunctor {
+  template <typename DeviceContext, typename X, typename Y, typename DX,
+            typename DY, typename Dim>
+  void operator()(const DeviceContext& place, X* x, Y* y, DX* dx, DY* dy,
+                  const Dim& dim, int size) {
+    dx->device(place) = y->broadcast(dim);
+    dx->device(place) = *dx + dx->constant(1e-12f);
+    dx->device(place) = (*x / *dx) * (dy->broadcast(dim));
+  }
+};
+
+}  // namespace operators
+}  // namespace paddle
diff --git a/python/paddle/__init__.py b/python/paddle/__init__.py
index 5587fd795b106838baeb3d85804713c51c777a5f..9e2bf8c5eea626f78c18b473db6f88a1ba8b7fd2 100644
--- a/python/paddle/__init__.py
+++ b/python/paddle/__init__.py
@@ -150,7 +150,7 @@ from .tensor.math import addmm  #DEFINE_ALIAS
 from .tensor.linalg import matmul  #DEFINE_ALIAS
 from .tensor.linalg import dot  #DEFINE_ALIAS
 # from .tensor.linalg import einsum   #DEFINE_ALIAS
-# from .tensor.linalg import morm   #DEFINE_ALIAS
+from .tensor.linalg import norm  #DEFINE_ALIAS
 # from .tensor.linalg import transpose   #DEFINE_ALIAS
 from .tensor.linalg import dist  #DEFINE_ALIAS
 # from .tensor.linalg import t   #DEFINE_ALIAS
diff --git a/python/paddle/fluid/tests/unittests/test_norm_all.py b/python/paddle/fluid/tests/unittests/test_norm_all.py
new file mode 100644
index 0000000000000000000000000000000000000000..e6b7a3e7603f53d78052d5de309d6ed7d84c4660
--- /dev/null
+++ b/python/paddle/fluid/tests/unittests/test_norm_all.py
@@ -0,0 +1,210 @@
+# Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from __future__ import print_function
+
+import unittest
+import numpy as np
+from op_test import OpTest
+import paddle
+import paddle.fluid as fluid
+
+
+def p_norm(x, axis, porder, keepdims=False):
+    if axis is None: axis = -1
+    xp = np.power(np.abs(x), porder)
+    s = np.sum(xp, axis=axis, keepdims=keepdims)
+    r = np.power(s, 1.0 / porder)
+    return r
+
+
+def frobenius_norm(x, axis=None, keepdims=False):
+    if isinstance(axis, list): axis = tuple(axis)
+    if axis is None: axis = (-2, -1)
+    r = np.linalg.norm(x, ord='fro', axis=axis, keepdims=keepdims)
+    return r
+
+
+class TestFrobeniusNormOp(OpTest):
+    def setUp(self):
+        self.op_type = "frobenius_norm"
+        self.init_test_case()
+        x = (np.random.random(self.shape) + 1.0).astype(self.dtype)
+        norm = frobenius_norm(x, self.axis, self.keepdim)
+        self.reduce_all = (len(self.axis) == len(self.shape))
+        self.inputs = {'X': x}
+        self.attrs = {
+            'dim': list(self.axis),
+            'keep_dim': self.keepdim,
+            'reduce_all': self.reduce_all
+        }
+        self.outputs = {'Out': norm}
+
+    def test_check_output(self):
+        self.check_output()
+
+    def test_check_grad(self):
+        self.check_grad(['X'], 'Out')
+
+    def init_test_case(self):
+        self.shape = [2, 3, 4, 5]
+        self.axis = (1, 2)
+        self.keepdim = False
+        self.dtype = "float64"
+
+
+class TestFrobeniusNormOp2(TestFrobeniusNormOp):
+    def init_test_case(self):
+        self.shape = [5, 5, 5]
+        self.axis = (0, 1)
+        self.keepdim = True
+        self.dtype = "float32"
+
+    def test_check_grad(self):
+        self.check_grad(['X'], 'Out')
+
+
+class TestPnormOp(OpTest):
+    def setUp(self):
+        self.op_type = "p_norm"
+        self.init_test_case()
+        x = (np.random.random(self.shape) + 0.5).astype(self.dtype)
+        norm = p_norm(x, self.axis, self.porder, self.keepdim)
+        self.inputs = {'X': x}
+        self.attrs = {
+            'epsilon': self.epsilon,
+            'axis': self.axis,
+            'keepdim': self.keepdim,
+            'porder': float(self.porder)
+        }
+        self.outputs = {'Out': norm}
+
+    def test_check_output(self):
+        self.check_output()
+
+    def test_check_grad(self):
+        self.check_grad(['X'], 'Out')
+
+    def init_test_case(self):
+        self.shape = [2, 3, 4, 5]
+        self.axis = 1
+        self.epsilon = 1e-12
+        self.porder = 2.0
+        self.keepdim = False
+        self.dtype = "float64"
+
+
+class TestPnormOp2(TestPnormOp):
+    def init_test_case(self):
+        self.shape = [3, 20, 3]
+        self.axis = 2
+        self.epsilon = 1e-12
+        self.porder = 2.0
+        self.keepdim = True
+        self.dtype = "float32"
+
+    def test_check_grad(self):
+        self.check_grad(['X'], 'Out')
+
+
+def run_out(self, p, axis, shape_x, shape_y, dtype):
+    with fluid.program_guard(fluid.Program()):
+        data1 = fluid.data(name="X", shape=shape_x, dtype=dtype)
+        data2 = fluid.data(name="Y", shape=shape_y, dtype=dtype)
+        out = paddle.norm(input=data1, p=p, axis=axis, out=data2)
+        place = fluid.CPUPlace()
+        exe = fluid.Executor(place)
+        result = exe.run(feed={"X": np.random.rand(*shape_x).astype(dtype)},
+                         fetch_list=[data2, out])
+        self.assertEqual((result[0] == result[1]).all(), True)
+
+
+def run_fro(self, p, axis, shape_x, dtype):
+    with fluid.program_guard(fluid.Program()):
+        data = fluid.data(name="X", shape=shape_x, dtype=dtype)
+        out = paddle.norm(input=data, p=p, axis=axis)
+        place = fluid.CPUPlace()
+        exe = fluid.Executor(place)
+        np_input = (np.random.rand(*shape_x) + 1.0).astype(dtype)
+        expected_result = frobenius_norm(np_input, axis=axis)
+        result, = exe.run(feed={"X": np_input}, fetch_list=[out])
+    self.assertEqual((np.abs(result - expected_result) < 1e-6).all(), True)
+
+
+def run_pnorm(self, p, axis, shape_x, dtype):
+    with fluid.program_guard(fluid.Program()):
+        data = fluid.data(name="X", shape=shape_x, dtype=dtype)
+        out = paddle.norm(input=data, p=p, axis=axis)
+        place = fluid.CPUPlace()
+        exe = fluid.Executor(place)
+        np_input = (np.random.rand(*shape_x) + 1.0).astype(dtype)
+        expected_result = p_norm(np_input, porder=p, axis=axis).astype(dtype)
+        result, = exe.run(feed={"X": np_input}, fetch_list=[out])
+    self.assertEqual((np.abs(result - expected_result) < 1e-6).all(), True)
+
+
+class API_NormTest(unittest.TestCase):
+    def test_output_result(self):
+        run_out(self, p=2, axis=1, shape_x=[3, 4], shape_y=[3], dtype="float32")
+        run_out(
+            self,
+            p='fro',
+            axis=None,
+            shape_x=[3, 4],
+            shape_y=[1],
+            dtype="float32")
+
+    def test_basic(self):
+        run_fro(self, p='fro', axis=None, shape_x=[3, 3, 4], dtype="float32")
+        run_fro(self, p='fro', axis=[0, 1], shape_x=[3, 3, 4], dtype="float64")
+        run_pnorm(self, p=2, axis=None, shape_x=[3, 4], dtype="float32")
+        run_pnorm(self, p=2, axis=1, shape_x=[3, 4], dtype="float64")
+
+    def test_name(self):
+        with fluid.program_guard(fluid.Program()):
+            x = fluid.data(name="x", shape=[10, 10], dtype="float32")
+            y_1 = paddle.norm(x, p='fro', name='frobenius_name')
+            y_2 = paddle.norm(x, p=2, name='pnorm_name')
+            self.assertEqual(('frobenius_name' in y_1.name), True)
+            self.assertEqual(('pnorm_name' in y_2.name), True)
+
+    def test_errors(self):
+        with fluid.program_guard(fluid.Program(), fluid.Program()):
+
+            def err_dtype(p, shape_x, xdtype, out=None):
+                data = fluid.data(shape=shape_x, dtype=xdtype)
+                paddle.norm(data, p=p, out=out)
+
+            self.assertRaises(TypeError, err_dtype, "fro", [2, 2], "int64")
+            out = fluid.data(name="out", shape=[1], dtype="int64")
+            self.assertRaises(TypeError, err_dtype, "fro", [2, 2], "float64",
+                              out)
+            self.assertRaises(TypeError, err_dtype, 2, [10], "int64")
+            self.assertRaises(TypeError, err_dtype, 2, [10], "float64", out)
+
+            data = fluid.data(name="data_2d", shape=[2, 2], dtype="float64")
+            self.assertRaises(ValueError, paddle.norm, data, p="unsupport norm")
+            self.assertRaises(ValueError, paddle.norm, data, p=[1])
+            self.assertRaises(ValueError, paddle.norm, data, p=[1], axis=-1)
+            self.assertRaises(
+                ValueError, paddle.norm, data, p='unspport', axis=[-2, -1])
+            data = fluid.data(name="data_3d", shape=[2, 2, 2], dtype="float64")
+            self.assertRaises(
+                ValueError, paddle.norm, data, p='unspport', axis=[-2, -1])
+            self.assertRaises(
+                ValueError, paddle.norm, data, p='unspport', axis=[-3, -2, -1])
+
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/python/paddle/tensor/__init__.py b/python/paddle/tensor/__init__.py
index 15a1607ac875477f407626121a9e300ff8d31ffd..0f1accd51ae1cf0a71dd13ec3c0f888f62fa3801 100644
--- a/python/paddle/tensor/__init__.py
+++ b/python/paddle/tensor/__init__.py
@@ -125,7 +125,7 @@ from .math import addmm  #DEFINE_ALIAS
 from .linalg import matmul  #DEFINE_ALIAS
 from .linalg import dot  #DEFINE_ALIAS
 # from .linalg import einsum   #DEFINE_ALIAS
-# from .linalg import morm   #DEFINE_ALIAS
+from .linalg import norm  #DEFINE_ALIAS
 # from .linalg import transpose   #DEFINE_ALIAS
 from .linalg import dist  #DEFINE_ALIAS
 # from .linalg import t   #DEFINE_ALIAS
diff --git a/python/paddle/tensor/linalg.py b/python/paddle/tensor/linalg.py
index 4d0d99edf4272534dace4ce40128aa40a390222c..d23c474d3077322695a99ed3359bb01ccc6d8d48 100644
--- a/python/paddle/tensor/linalg.py
+++ b/python/paddle/tensor/linalg.py
@@ -20,7 +20,7 @@ __all__ = [
     'matmul',
     'dot',
     #  'einsum',
-    #  'morm',
+    'norm',
     #  'transpose',
     'dist',
     #  't',
@@ -160,6 +160,181 @@ def matmul(x, y, transpose_x=False, transpose_y=False, alpha=1.0, name=None):
     return out
 
 
+def norm(input, p='fro', axis=None, keepdim=False, out=None, name=None):
+    """
+    Returns the matrix norm (Frobenius) or vector norm (the 1-norm, the Euclidean
+    or 2-norm, and in general the p-norm for p > 0) of a given tensor.
+
+    Args:
+        input (Variable): The input tensor could be N-D tensor, and the input data
+            type could be float32 or float64.
+        p (float|string, optional): Order of the norm. Supported values are `fro`, `1`, `2`,
+            and any positive real number yielding the corresponding p-norm.
+        axis (int|list, optional): The axis on which to apply norm operation. If axis is int
+            or list with only one element, the vector norm is computed over the axis.
+            If axis is a list with two elements, the matrix norm is computed over the axis.
+            If `axis < 0`, the dimension to norm operation is rank(input) + axis.
+        keepdim (bool, optional): Whether to reserve the reduced dimension in the
+            output Tensor. The result tensor will have fewer dimension
+            than the :attr:`input` unless :attr:`keepdim` is true, default
+            value is False.
+        out (Variable, optional): The output tensor, default value is None. It's data type
+            must be the same as the input Tensor.
+        name (str, optional): The default value is None. Normally there is no need for
+            user to set this property. For more information, please refer to :ref:`api_guide_Name`.
+
+    Returns:
+        Variable: Tensor, results of norm operation on the specified axis of input tensor,
+        it's data type is the same as input's Tensor.
+ 
+    Raises:
+        TypeError, if out data type is different with the input data type.
+        ValueError, If `p` or `axis` is invalid.
+    
+    Examples:
+        .. code-block:: python
+            
+            import paddle
+            import paddle.fluid as fluid
+            x = fluid.data(name='x', shape=[2, 3, 5], dtype='float64')
+            
+            # compute frobenius norm along last two dimensions.
+            out_fro = paddle.norm(x, p='fro', axis=[1,2])
+            
+            # compute 2-order vector norm along last dimension.
+            out_pnorm = paddle.norm(x, p=2, axis=-1)
+    """
+
+    def frobenius_norm(input, dim=None, keepdim=False, out=None, name=None):
+        """
+        The frobenius norm OP is to calculate the frobenius norm of certain two dimensions of Tensor `input`.
+        Args:
+          input (Variable): Tensor, data type float32, float64.
+          dim (list, optional): None for last two dimensions.
+          keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False.
+          out (Variable, optional): The tensor variable storing the output.
+        """
+        if dim is not None and not (isinstance(dim, list) and len(dim) == 2):
+            raise ValueError(
+                "The dim of frobenius norm op should be None or two elements list!"
+            )
+        attrs = {
+            'dim': dim if dim != None else [-2, -1],
+            'keep_dim': keepdim,
+            'reduce_all': False
+        }
+        if len(attrs['dim']) == len(input.shape):
+            attrs['reduce_all'] = True
+        check_variable_and_dtype(input, 'input', ['float32', 'float64'],
+                                 'frobenius_norm')
+
+        helper = LayerHelper('frobenius_norm', **locals())
+        if out is None:
+            out = helper.create_variable_for_type_inference(
+                dtype=helper.input_dtype())
+        else:
+            check_type(out, 'out', (Variable), 'frobenius_norm')
+            check_dtype(
+                out.dtype, out.name,
+                convert_dtype(input.dtype), 'frobenius_norm',
+                '(The out data type in frobenius_norm must be the same with input data type.)'
+            )
+
+        helper.append_op(
+            type='frobenius_norm',
+            inputs={'X': input},
+            outputs={'Out': out},
+            attrs=attrs)
+        return out
+
+    def vector_norm(input,
+                    porder=None,
+                    axis=None,
+                    keepdim=False,
+                    out=None,
+                    name=None):
+        """
+        Calculate the p-order vector norm for certain  dimension of Tensor `input`.
+        Args:
+          input (Variable): Tensor, data type float32, float64.
+          porder (float, optional): None for porder=2.0.
+          axis (int, optional): None for last dimension.
+          keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False.
+          out (Variable, optional): The tensor variable storing the output.
+        """
+        if porder is not None:
+            check_type(porder, 'porder', (float, int), 'p_norm')
+        if axis is not None:
+            check_type(axis, 'axis', (int), 'p_norm')
+        attrs = {
+            'axis': axis if axis is not None else -1,
+            'porder': float(porder) if porder is not None else 2.0,
+            'keepdim': keepdim,
+            'epsilon': 1e-12,
+        }
+        check_variable_and_dtype(input, 'input', ['float32', 'float64'],
+                                 'p_norm')
+
+        helper = LayerHelper('p_norm', **locals())
+        if out is None:
+            out = helper.create_variable_for_type_inference(
+                dtype=helper.input_dtype())
+        else:
+            check_type(out, 'out', (Variable), 'p_norm')
+            check_dtype(
+                out.dtype, out.name,
+                convert_dtype(input.dtype), 'p_norm',
+                '(The out data type in p_norm must be the same with input data type.)'
+            )
+
+        helper.append_op(
+            type='p_norm',
+            inputs={'X': input},
+            outputs={'Out': out},
+            attrs=attrs)
+        return out
+
+    if axis is None and p is not None:
+        if isinstance(p, str):
+            if p == "fro":
+                return frobenius_norm(
+                    input, dim=axis, keepdim=keepdim, out=out, name=name)
+            else:
+                raise ValueError(
+                    "only valid string values are 'fro', found {}".format(p))
+        elif isinstance(p, (int, float)):
+            return vector_norm(
+                input, porder=p, axis=axis, keepdim=keepdim, out=out, name=name)
+        else:
+            raise ValueError("only valid p type is string or float, found {}".
+                             format(type(p)))
+
+    if isinstance(axis, list) and len(axis) == 1:
+        axis = axis[0]
+
+    #calculate vector norm, where axis is int or list with only one integer
+    if isinstance(axis, int):
+        if isinstance(p, (int, float)):
+            return vector_norm(
+                input, axis=axis, porder=p, keepdim=keepdim, out=out, name=name)
+        else:
+            raise ValueError(
+                "unspport p for p-order vector norm. except float, found {}".
+                format(p))
+    #calculate matrix norm, where axis is list with two integers
+    elif isinstance(axis, list) and len(axis) == 2:
+        if p == "fro":
+            return frobenius_norm(
+                input, dim=axis, keepdim=keepdim, out=out, name=name)
+        else:
+            raise ValueError(
+                "unspport p for matrix norm, expcept 'fro', found {}".format(p))
+    else:
+        raise ValueError(
+            "except axis type int or list (length of list <=2), found {}".
+            format(axis))
+
+
 def dist(x, y, p=2):
     """
     This OP returns the p-norm of (x - y). It is not a norm in a strict sense, only as a measure